CAPÍTULO I – Matemática Básica Expressões Numéricas · Prof. Cícero José – Anhanguera Uniban 2012 1 CAPÍTULO I – Matemática Básica Expressões Numéricas 1) Calcule

Prof. Cícero José – Anhanguera Uniban 2012

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CAPÍTULO I – Matemática Básica

Expressões Numéricas 1) Calcule o valor das expressões abaixo: a) 20 – [(8 – 3) + 4] – 1 b) 123 – [90 – (38 + 50) – 1]

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c) 10 + [–8 – (–1 + 2)] d) –3 – [8 + (–6 – 3) + 1]

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e) 8 – (4 + 5) – [3 – (6 – 11)] f) –(–2) – [9 + (7 – 3 – 6) – 8]

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g) 1 + [–7 – (–2 + 6) + (–2)] – (–6 + 4) h) 6 – {4 + [–7 – (–3 – 9 + 10)]}

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i) –3 – [(–1 + 6) + 4 – (–1 – 2) – 1] j) 2 – (–2) – {–6 – [–3 + (–3 + 5)]} – 8

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2) Calcule o valor das expressões abaixo: a) 21 – 15 : 5 – 12 + 3 + 1 b) (21 – 15) : (15 – 12 + 3) + 1

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c) 31 – 40 : 2 d) –10 – 20 : 4

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e) 30 : (–6) + (–18) : 3 f) 7 : (–7) + 2(–6) + 11

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3) Escreva a expressão numérica que representa cada situação abaixo: a) Um milionário, antes de morrer, deixou escrito no testamento: “Dos três milhões que tenho no

banco, deixo 1 milhão e 800 mil para instituições de caridade e o restante para ser repartido igualmente

entre meus três filhos”. Quanto recebeu cada filho?

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b) João tem 26 tickets refeição e André tem o triplo. Quantos tickets refeição têm os dois juntos?

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c) Dois operários, Paulo e Pedro, cobram juntos, R$ 385,00 por um trabalho a ser realizado em 5

dias. Paulo ganha R$ 32,00 por dia de trabalho. Quanto ganhou Pedro pelo trabalho?

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d) Gaspar comprou uma bicicleta pagando um total de R$ 960,00, sendo R$ 336,00 de entrada e o

restante em 8 prestações mensais iguais. Qual o valor de cada prestação?

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e) Em cada mão humana há 27 ossos e em cada pé, 26. Quantos ossos há, ao todo, nas mãos e nos

pés humanos?

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f) José mandou fazer, de alumínio, as janelas de sua casa. Deu uma entrada de R$ 250, 00 quando

fez a encomenda e o restante vai pagar em quatro parcelas iguais de R$ 140,00 cada uma. Qual a

quantia que José vai gastar para fazer as janelas?

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g) O preço de uma corrida de táxi é formado de duas partes: uma fixa, chamada “bandeirada”, e uma

variável, de acordo com o número de quilômetros percorridos. Em uma cidade, a “bandeirada” é de

R$ 4,00 e o preço por quilômetro percorrido é de R$ 2,00. Quanto pagará uma pessoa que percorrer, de

táxi, 12 quilômetros?

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h) Regina comprou roupas, gastando um total de R$ 814,00. Deu R$ 94,00 de entrada e o restante da

dívida vai pagar em 5 prestações mensais iguais. Qual é o valor de cada prestação?

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CAPÍTULO II – Cálculo Algébrico

Parte I – Monômios 4) Determine as seguintes somas algébricas: a) –5a + 3a b) xy + xy

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c) –ac – 5ac d) 10am – 13am

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e) –3a2 + 4a2 f) –xy2 + 7xy2

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g) 2bc – 15

bc h) 2 21 2x x

2 5−

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i) 3

mn 2mn4

− j) 3x – 10x + 11x

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k) –2y2 + 3y2 – 5y2 l) 6ab – 11ab + 6ab

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m) 5a2m – 12a2m + 7a2m n) –xy + 3xy + 4xy – 2xy

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o) –10n3 + 8 n3 – 7n3 + 12n3 p) –5am + 8am – 3am + am – 6am

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q) a4 + 4 42 3a a

3 2− r)

1 4 1bc bc bc

2 5 10− −

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____________________________________ ____________________________________ 5) Reduzindo os termos semelhantes, simplifique as expressões algébricas: a) 2y3 – 7y + y3 + 5y – y b) 5a – 10ab + 4b – 4a + 8ab

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c) 6x2 – 8x + 3x2 – 5 + 10x + 4 d) mn + 3m – 5n + 4mn – m + 6n – 2mn

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e) 2a2 – 5ab + 7b2 + 4ab – a2 + 2b2 f) x + y – 2 + 3x + 5 – 2y – x + 1 – y

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g) 1 2

a + b + a 2b2 3

− h) 2 21 1 1x + x + x + 3x x

2 4 8−

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6) Sabemos que um triângulo é equilátero quando todos os seus lados têm a mesma medida. Se você

representar a medida do lado do triângulo pela letra x, como poderá representar, de forma simbólica, o

perímetro desse triângulo?

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7) Escreva a expressão algébrica que representa cada situação abaixo: a) a soma do quadrado do número x com o quíntuplo do número y.

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b) a soma dos quadrados dos números x e y.

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c) o quadrado da soma dos números x e y.

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d) o produto da soma de a e b pela diferença desses dois números.

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e) o perímetro do retângulo de base a e altura h.

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f) a soma dos cubos dos números a e b.

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g) o cubo da soma dos números a e b.

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h) a diferença entre os quadrados dos números x e y.

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i) a terça parte do quadrado do número x.

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j) a diferença entre o número x e 5.

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8) Com vistas à reforma agrária, uma fazenda foi desapropriada pelo Governo Federal e dividida em

100 lotes, todos de forma quadrada e de mesma área, para distribuição entre os “sem-terra”. Determine

a expressão algébrica que expressa a área A do terreno em função da medida x do lado de cada lote.

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9) Duas lojas vendem o mesmo artigo pelo mesmo preço x para pagamento à vista. Para compra a

prazo, esse artigo tem preços diferentes:

Loja 1: entrada de 40% do preço x mais três prestações iguais de y reais.

Loja 2: entrada de 30% do preço x mais duas prestações iguais de y reais. Nessas condições, escreva o polinômio que expressa:

a) O preço do artigo comprado a prazo na loja 1.

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b) O preço do artigo comprado a prazo na loja 2.

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c) A diferença entre o preço na loja 1 e o preço na loja 2.

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10) Pedro é estagiário em uma empresa. Ele recebe R$ 5,87 a hora. No mês de agosto ele trabalhou 157

horas. Determine a expressão numérica que representa seu salário.

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11) Calcule o valor numérico das expressões abaixo:

a) 2a + 3b, para a = –2 e b = –3 b) x2 + 2x, para x = –5

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c) x + yx y−

, para x = 4 e y = –2 d) x y

+ 3 4

, para x = 9 e y = –8

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e) (x – y)2, para x = 9 e y = –3 f) (x + y)2, para x = 5 e y = –9

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12) Elimine os parênteses, os colchetes e as chaves e reduza os termos semelhantes. a) x – (–2y + 3x) + (5x – 4y)

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b) a2 – (–2a + 5) + a – (–3a2 + 4a – 1)

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c) 10x3 – (x2 + 3x – 1) + (x3 – 3x2 + 2) – (4x – 3)

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d) 2a + [–5b + 2c – (a + 2b – c)] – (4b – 2c)

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e) x2 – [2xy + x2 – (y2 + 3xy) + 2y2] – xy

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f) ab – {–bc – [ac + (ab – ac – bc) + bc]}

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g) (a – x) + [(2a – x) – 7] – (a – 2x – 6)

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h) 2a + [(a – b) + (c – d)] + (b – c)

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i) 5x – {[3x – (7 – 5y)] + [x – (9 – 6y)] – (2x + y)}

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j) {(3x – y) – [(x – y) – (3x – 5y)]} – (2x – 4y)

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13) Determine os seguintes produtos: a) (–3y) � (+9y) b) (–2xy) � (–5x)

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c) (6b) � (4c) d) (x2y3) � (–xy2)

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e) (10xy3) � (x3) f) (4m2nx) � (–3mx2)

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g) 41 3x y xy

2 5� � � �−� � � ��

� h) ( )25p q 2pq

8� �− −� ��

�

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i) (–5x2) � (2x) � (–x3) j) (ax) � (–6a2) � (3x)

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k) (–2a) � (–5b) � (–6ab3) l) (–mx2) � (xy2) � (mxy2)

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m) 2 21 1xy x y

2 3� � � ��

� � (–6xy) n) 22 1hx h

3 5� � � �− −� � � ��

� � (–hxy)

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o) (–kx2) � (–2kx) � (–5x) � (3) p) 1 1

3 anp 2 bnx3 4

� � � �−� � � ��

�

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14) Determine os seguintes quocientes: a) (–24y5) : (–6y2) b) (5x3) : (x) c) (–18a2) : (6a2)

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d) (–2x2) : (–2x) e) (6a4) : (–6a4) f) (10xy) : (5x)

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g) (a4b2c) : (–abc) h) (–4m2n2) : (–mn2) i) (–a4b2) : (2a3b)

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j) 5 41 1x : x

3 5� � � ��

k) 32 4mn : mn

3 3� � � �−� � � ��

l) ( )1ab : 2b

4� �− −� ��

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m) (2a3b5c2) : 1

abc3

� �−� ��

n) (a4b4c) : (a3b2)

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15) Determine as seguintes potências: a) (–4x3y)2 b) (–mx2)3 c) (2ac3)5 d) (–3b2c)4

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e) (–h2m)6 f)

341

y2

� �−� ��

g) 2

2mn

5� �−� ��

h) (x2y5)5

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i) (–2a3)10 j) (–4a3c)0 k) 6

31a

2� �−� ��

l) (–0,5x4y7)2

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Parte II – Polinômios

Polinômios 16) Dados A = 2m2 + 5m + 3, B = 4m2 – 2m + 1 e C = –3m2 – m + 3, determine: a) A – B

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b) B – A

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c) A – C

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d) B – C

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e) C – A

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f) C – B

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17) Calcule:

a) 1 1 3

y c c a2 2 4

� � � �− − −� � � ��

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b) 4 2 4 23 1x x + x x +

5 2� � � �− − −� � � ��

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c) 3 31 15m m + + m 7m

4 2� � � �− −� � � ��

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d) 2 21 1 3x 2x + 1 + x +

2 2 2� � � �−� � � ��

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18) O número de cada retângulo é obtido adicionando os números dos dois retângulos situados abaixo.

Escreva uma expressão simplificada no retângulo colorido superior.

a) b) c)

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19) Calcule os seguintes produtos: a) 5x � (ax2 + bx + c)

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b) –3xy2 � (5x + 6y – 7xy)

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3 x 2x

3 + x

–2x –7 –x

–7 – x

x 3x

1 + x

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c) mn � (m2 – mn + n2)

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d) (3a – 4b + 5c) � 2x

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e) (–a + 3b – 5c) � (–7ax)

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f) (3x2 – 2x + 1) � 2x

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g) 3x2

� (8x2 + 6x + 5)

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20) Dados os polinômios A = x – 5, B = x2 – 7x + 12 e C = x2 – 6, determine: a) A � B b) A � C

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c) B � C d) (A + B) � C

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e) (C – A) � B f) (A + B – C) � B

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21) (Saresp-SP) Qual expressão algébrica representa a área da figura? _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________

a

a a

a

a2

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22) Calcule os quocientes: a) (12x2 + 9x) : 3x b) (–6x2 + 4x) : 2x

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c) (x4 + 5x3 + x2 – 4x) : x d) (–8a4 + 6a3 – 10a) : (–2a)

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e) (–10m4 + 35m3 – 15m2) : (–5m) f) (40x2 – 20x – 3ax) : (–10x)

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23) Efetue as divisões de polinômios: a) (x2 + 9x + 14) : (x + 7) b) (6x2 – 13x + 8) : (3x – 2)

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c) (12x2 – 11x – 15) : (4x + 3) d) (2x3 – 5x2 + 6x – 4) : (x – 1)

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e) (–15x3 + 29x2 – 33x + 28) : (3x – 4) f) (x3 – 6x2 – x + 30) : (x2 – x – 6)

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g) (2x4 + 3x3 – x2 + 7x – 3) : (2x2 – x + 3) h) (–8x4 – 8x3 + 6x2 – 16x + 8) : (4x3 + 6x2 + 8)

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i) (3x3 – 30x + 2) : (x2 – 3x + 1) j) (x4 + x2 – 3x + 1) : (x2 – x – 1)

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k) (8x4 – 6x2 + 3x – 2) : (2x2 – 3x + 2) l) (x3 – 64) : (x – 4)

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24) Escreva o polinômio que permite calcular a área da parte colorida da figura. _________________________________________________________________________________

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3x 5 x x 4

2x


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25) (Saresp-SP) Numa padaria há um cartaz afixado em que constam os seguintes itens:

LEITE R$ 0,70 PÃO R$ 0,12

Joana comprou uma quantidade x de litros de leite e uma quantidade y de pães. Determine a expressão

algébrica que representa essa compra.

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26) O tangran é um jogo chinês de formas, uma espécie de

quebracabeças, que consta de sete peças com as quais se podem compor

numerosas figuras. Na foto, as sete peças formam um quadrado.

Determina a área de cada peça, sabendo-se que a soma de todas as áreas

corresponde a 4x2.

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Brincando... com a álgebra 27) Pense em um número inteiro de 20 a 29. Some os algarismos do número. Agora, subtraia essa

soma do número pensado.

a) Qual vai ser o resultado?

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b) Imaginando que o número pensado seja 20 + x, efetue os cálculos algébricos para mostrar que o

resultado é sempre o mesmo.

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1 2 3

4 5

6

7

x

x

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28) No exercício anterior, se o número pensado for um número inteiro de 30 a 39, qual será o resultado

final?

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29) Se você escrever um número natural, de 40 a 49, multiplicá-lo por 10 e subtrair 396, você obterá o

mesmo número, mas com a ordem dos algarismos invertida.

a) Faça esse truque com o número 46. Qual será o resultado?

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b) Utilizando a álgebra, explique por que esse truque sempre dá certo.

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30) Com três números naturais consecutivos acontece uma pequena surpresa. Multiplicando o menor

pelo maior e depois somando 1, obtém-se o número do meio, elevado ao quadrado.

a) Mostre que isso acontece com os números 6, 7 e 8.

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b) Mostre que isso acontece sempre, utilizando os números x – 1, x e x + 1.

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31) Veja o desafio da professora:

• Pense em número natural. Some 3. Multiplique o resultado pelo número pensado.

• Agora, some 2. Divida o resultado pelo sucessor do número pensado.

• No final, deu o número pensado, mais 2, não é? Mostre que isso sempre acontece, usando

álgebra.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

32) Efetue a seguinte sequência de cálculos: Pense em um número real x, não-nulo. Some 7.

Multiplique o resultado por 5. Subtraia 35. Divida o resultado pelo número pensado. Qual é o resultado

final?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

33) Pense em um número natural qualquer. Some 5. Multiplique por 3. Subtraia 12. Divida por 3. O

resultado é o sucessor do número que você pensou. Usando álgebra, explique por que isso sempre

acontece.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


25

34) Adivinhando data de aniversário: – Multiplique o número do mês por 5 e adicione 7.

– Multiplique por 4 e adicione 13.

– Multiplique por 5.

– Adicione o dia do mês correspondente ao seu aniversário.

– Agora subtraia 205 da resposta. Os dois primeiros números representam o mês do seu aniversário e dois últimos representam o dia do

seu nascimento. Usando álgebra, explique por que isso sempre acontece.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

35) – Pense em um número inteiro de 10 a 19, mas não me diga qual é. Some os dois algarismos.

– Agora, subtraia essa soma do número que você pensou.

– Agora, eu vou adivinhar o resultado que você encontrou. O resultado é nove! Certo?

Usando álgebra, explique por que isso sempre acontece.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


26

CAPÍTULO III – Produtos Notáveis

36) Desenvolva os seguintes produtos notáveis: a) (x2 + y3)2 b) (5ax2 + 6a3)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) (4a3x + 2by)2 d) (a2 + 5am)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

e) (x6 + 3x2)2 f)

25x 2y

+ 3 3

� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

____________________________________ ___________________________________

____________________________________ ___________________________________

g) 2

4 13x +

3� ��

h) 223a

+ a4

� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

____________________________________ ___________________________________

____________________________________ ___________________________________ ____________________________________ ___________________________________

i) 23 4x x

+ 3 4

� ��

j) 22 2ab a b

+ 3 3

� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

____________________________________ ___________________________________

____________________________________ ___________________________________ ____________________________________ ___________________________________


27

k) (a3 – 4a)2 l) (3x2 – 5xy)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

m) (5a2 – 4b2)2 n) (x3 – y3)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

o) (2ax2 – 4a2x2) p)

23 4

x3

� �−� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

____________________________________ ___________________________________

____________________________________ ___________________________________ ____________________________________ ___________________________________

q) 2

2 32 1a b a x

3 2� �−� ��

r) 22 2m x 3y

4 5� �

−� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

____________________________________ ___________________________________

____________________________________ ___________________________________ ____________________________________ ___________________________________

s) 232a x 5by

3 2� �

−� ��

t) 26x 5

+ 2 4

� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________


28

37) Desenvolva os seguintes produtos notáveis: a) (a2 – 1) (a2 + 1) b) (a3b2 + c) (a3b2 – c)

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) (x3 + y3) (x3 – y3) d) (5a3b + 2xy2) ((5a3b – 2xy2)

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

e) (x2y4 – 5x) (x2y4 + 5x) f)

2 2x 3 x 3 +

5 4 5 4� ��

−� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________

g) 2 23m 1 3m 1

+ 2 3 2 3

� �� −� ��

� �� h) 3 3b b

a + a5 5

� �� −� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________

i) 2 25x 5x

+ 6 63 3

� �� −� ��

� �� j)

4 4x x + 6 6 +

3 3� ��

−� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________


29

38) Desenvolva os seguintes produtos notáveis: a) (a2 + b2)3 b) (2ax + 3a2)3

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) (x3 + x)3

d) 3

3x 1 +

2 3� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________

e) 32a b

+ 4 3

� ��

f) 32ax bx

+ 2 3

� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ g) (x2 – y3)3 h) (3a2 – b3x)3

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

i) (m2 – 3xy2)3

j) 22x 3

3 4� �

−� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________


30

k) 331 a

2 4� �

−� ��

l) 32x y

2 3� �

−� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ 39) Desenvolva os seguintes produtos notáveis: a) (x + y + 3)2 b) (x2 + y + 1)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) (2x – y – 1)2 d) (x – 4y + 3)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

e) (m5 – 1)2 f)

21

8xy4

� �−� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________

g) (x + 5)3 h) (x – c)3

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________


31

i) (5x5 + 2x)2 j) 2 21 1

5m + 5m2 2

� �� −� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ k) (5m3x – 7n3z2)(5m3x + 7n3z2) l) (3a + 2y)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

m) 2

2 33m 5n

2� �−� ��

n) 2 24 2 4 2a b a b +

5 9 5 9� �� −� ��

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________

o) 3

1xy 3x

3� �−� ��

p) 2

2 12x + y

2� ��

____________________________________

___________________________________

____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ 40) Calcule os seguintes produtos, utilizando o produto de Stevin: a) (x + 6)(x + 4) b) (a – 3)(a – 5) c) (y – 7)(y + 3)

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________


32

d) (m – 12)(m + 8) e) (x – 9)(x + 5) f) (x + 15)(x + 3)

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

g) (b – 4)(b + 20) h) (k – 8)(k – 13) i) (x + 11)(x – 6)

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

j) (u – 4)(u + 15)

k) 1 3

x + x + 2 4

� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

l) 3 5

x x + 5 3

� �� −� ��

m) ( )2x + x 9

3� � −� ��

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

n) 2 3

x x5 2

� �� − −� ��

o) ( )7x + x 2

3� � −� ��

p) 3 7

x + x4 4

� �� −� ��

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


33

41) Qual é o valor de x2 + y2, sabendo que x + y = 7 e xy = 10?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

42) Qual é o valor de 22

1x +

x, sabendo que

1x

x− = 9.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

43) Qual é o valor de x2 + y2, sabendo que x – y = 15 e xy = 100?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

44) Usando produto notável, calcule: a) 51 � 49 b) 202 � 198 c) 77 � 63 d) 21 � 19

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

45) Qual é o valor de 22

14x +

x, sabendo que

12x

x− = 20?

_________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________


34

46) Qual é o valor de x – y, sabendo que x2 – y2 = 800 e x + y = 100?

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

47) Sendo A = (x + 2)2, B = (x + 3)(x – 3) e C = (x – 1)2, determine o valor de A + B – C.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

48) Sabendo que 1

aa

− = 3, calcule o valor de 22

1a +

a.

_________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________

49) Mostre que a diferença entre os quadrados da soma e da diferença de dois números inteiros não

nulos é sempre divisível por cada um deles e pelo número quatro.

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


35

50) Sendo A = 2

1x +

x� ��

e B = 2

1x

x� �−� ��

, determine o valores de (A – B)2 e (A + B)2.

_________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ 51) Sendo S = ( )( )x + 7 x 7− , P = (x – 3)2 – 12 e Q = (x + 5)(x – 2)(x – 1), determine Q – (S + P).

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

52) Sendo 1

a + a

= 35

, determine o valor de 33

1a +

a.

_________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________

53) (Olimpíada Bras. de Matemática) Se x + y = 8 e xy = 15, qual é o valor de x2 + 6xy + y2?

a) 109 b) 120 c) 124 d) 154 _________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________

_________________________________________________________________________________


36

CAPÍTULO IV – Fatoração

54) Fatore os seguintes polinômios. a) a3 – ax b) 5ax – 5a3x2 c) 7p2 + p

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

d) 15a2 – 225a4 e) 15 + 25x2 f) 6x3 + 2x4 + 4x5

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

g) 2x2y3 – 6x2y2 + 2xy3 h) 3a4 – 3a3b + 6a2b2 i) 5x5 – 10a2x3 – 15a3x3

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

j) 6x3 – 9x2y + 12xy2

k) 5 2 4 2 21 1 1a b c + a b + a b

12 6 2−

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________

l) 5 4 5 4 6 3 71 1 1a m n m n + m n

20 20 8− m) 2 3 2 3 2 56 9 9

h k x h k y + h k15 20 10

−

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________


37

n) 7 4 3 7 5 92 1 2a b a b + a b

15 12 3− o) 2 10 6 9 2 46 12 18

y z + y z y z35 45 25

−

____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ___________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ ____________________________________ ___________________________________ 55) Fatore os seguintes polinômios. a) a2 + ab + ac + bc b) 2x + cx + 2c + c2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) 5a + ab + 5b + b2 d) mx – my – nx + ny

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

e) a2 – ac + ax – cx f) 3ax – bx – 3ay + by

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

g) 6x2 + 3xy – 2ax – ay h) ax2 – 3bxy – axy + 3by2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________


38

i) x2 – 3x – xy + 3y j) x4 + x3 + 2x + 2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

k) 3 21 1 1 1a + a + a +

3 5 3 5 l) 2 21 1

x + mxy xy my4 4

− −

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

m) 3

2m m 2 2m +

2 6 3 9− −

n) 6am + 4m + 15an + 10n

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

o) a5 – a2b + a3 – b p) 36am + 45an + 4m + 5n

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

56) Fatore os seguintes polinômios. a) r2 – x2 b) a2 – 4 c) m2 – 9 d) b2 – 16

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________


39

e) 25 – y2 f) 16 – x4 g) a4 – 100 h) n2p2 – 1

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

i) x4 – y4 j) 16x2 – 25y2 k) x6 – 144a4 l) x4 – a2b2

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

m) 9a4 – 16n2 n) 21

y4

− o) 2 2a b

9 16− p)

2 2

2 2

x ya b

−

________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________

q) 2 2 2 2m r x p

16 25−

r) (a – b)2 – (a + b)2

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

s) (1 + a)2 – (1 – a)2 t) (a – 2)2 – (a + 3)2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

57) Fatore os seguintes polinômios, quando possível. a) 4x2 + 4x + 1 b) x2 + 10x + 25 c) 4x4 + 4x2y3 + y6 _______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________


40

d) 16 – 8x + x2 e) 36m2n4 – 24mn2x3 + 4x6 f) 25x4y2 – 20x2yp3 + 4p6 _______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

g) 4x10 + 12x5y2m + 9y4m2 h) 4x2 + 1 – 4x i) x2 + a2 – 2ax _______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

j) 6 3 29 12x + x y + 4y

25 5 k) 8 41 9

m m + 9 4

−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

l) 6 3 4 2 8 41 1 1a + a b c + b c

16 4 4 m) 6 3 24 1

4x + x y + y3 9

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

n) 36m4y6 + 6m2y5x + 2 41x y

4

o) x2 + y8 – 2y4x

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

p) a2 + 4ab + 9b2 q) 4x2 – 4x + 1 r) a2 – ab + b2

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

s) 16x2 – 20xy + 9y2 t) a2 + b + 14

u) 21 2x + x + 1

9 3

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


41

58) Fatore os seguintes trinômios. a) x2 + 7x + 12 b) x2 + 7x + 10 c) x2 – 7x + 6 d) x2 – 6x + 8

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

e) x2 – 9x + 14 f) x2 + x – 12 g) x2 – 9x + 18 h) x2 – 9x + 8

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

i) x2 – x – 12 j) x2 + 4x – 12 k) x2 + 7x – 8 l) x2 – 2x – 15

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

m) x2 + x – 6 n) x2 – 5x + 4 o) y2 – 11y – 12 p) m2 – 13m + 12

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

q) t2 + 8t + 12 r) a2 – 2a – 8 s) k2 + 13k + 40 t) z2 – 7z – 8

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

59) Aplicando os casos de fatoração estudados, fatore os polinômios. a) x2 + 5x b) 4x2 – 12x + 9 c) x3 – 2x2 + 4x – 8

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

d) 4x2 – 9 e) a6 – 5a5 + 6a3 f) ax – a + bx – b

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________


42

g) 64y2 + 80y + 25 h) a3b2 + a2b3 i) m6 – 1

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

j) 4a2x2 – 4abx + b2 k) 12a2b + 18a l) x3 – x2y + xy – y2

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

m) (x + 1)2 – 9 n) a2bc + ab2c + abc2 o) 25x2 + 70x + 49

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

p) 1 – (a + b)2 q) x6 + x4 + x2 + 1 r) 15a3m – 20a2m

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

s) 21m

4– 25n2

t) 81y2 + 18y + 1

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

u) x2 + 3x + 2 v) m2 + 3m – 4

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

w) m2 – 2m – 3 x) x2 + 3x – 10 y) m2 – m – 2 z) x2 – 13x + 36

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________


43

60) Fatore os seguintes polinômios, usando sucessivamente os casos de fatoração. a) 10a2 – 10 b) x3 – 10x2 + 25x c) 2m2 – 8

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

d) ay2 + 4ay + 4a e) h4 – m4 f) x2y – 36y

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

g) ab2 – a + b2c – c h) x4 + 2x3 + x2 i) 81 – k4

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

j) x3y – 8x2y2 + 16xy3 k) x2y – 8xy + 16y l) 4z2 – 120z + 900

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

m) 5x2 – 20 n) 4x2z – 25z o) 12ax4 – 24a2x2 + 12a3

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

p) 4x3 – 8x2 + 4x q) a3 + a2 – 4a – 4 r) x6 – y6

_______________________ _______________________ _______________________

_______________________

_______________________

_______________________

_______________________

_______________________

_______________________

s) 3(a + b – c)2 – 3(a – b + c)2 t) x3 – 6x2 + 9x – x5

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________


44

61) (Furb–SC) Um professor de Matemática tem 4 filhos. Em uma de suas aulas, ele propôs aos alunos

que descobrissem o valor da expressao ac + ad + bc + bd, sendo que a, b, c e d são as idades de seus

filhos na ordem crescente. O professor disse, também, que a soma das idades dos dois mais velhos é 59

anos e a soma das idades dos dois mais novos é 34 anos. Qual o valor numérico da expressão proposta

pelo professor?

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

62) Efetue: a) 502 – 492 b) 20002 – 19992 c) 502 – 482

_______________________ _______________________ _______________________

_______________________

_______________________

_______________________

_______________________

_______________________

_______________________

d) 20012 – 20002 e)

2 2123 456 123 455123 456 + 123 455

−.

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

63) Multiplique um número natural pelo sucessor de seu sucessor. (Por exemplo, 3 . 5 ou 9 . 11). Some

1 ao resultado. Aí, extraia a raiz quadrada. Surpresa! Essa raiz quadrada é sempre um número inteiro.

Usando álgebra, explique por que isso acontece.

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________


45

64) A diferença dos quadrados de dois inteiros consecutivos pode ser um número par?

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

65) Determine o valor de 2

2 2

1 000252 248−

.

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

66) x e y são as medidas dos lados de um retângulo de área 20 e perímetro 18. Qual é o valor numérico

da expressão 3x2y + 3xy2?

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

67) Fatore as seguintes expressões: a) x9 + y6 b) m6 + 8n12 c) y12 – 27

_______________________ _______________________ _______________________

_______________________

_______________________

_______________________

_______________________

_______________________

_______________________

x

y


46

d) 8a3 – 27b3 e) 64 + a6 f) 125x6 – 1

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

g) x6y3 – a9b12

h) 9x 64

+ 8 27

i) 3 15a m

1125

−

_______________________ _______________________ _______________________ _______________________ _______________________

_______________________ _______________________

_______________________ _______________________

_______________________ _______________________ _______________________ _______________________ _______________________

_______________________ _______________________

_______________________ _______________________

j) 12 98x 216a

+ 27 125

k) 27a6 – 31b

8 =

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ 68) (SEE-RJ) O resultado de uma expressão é a2 – b2.

• Sílvio encontrou como resposta (a – b)2;

• Cláudio, (a + b)(a – b);

• Célia, (a + b)2 – 2b2 Como o professor aceita o desenvolvimento incompleto da resposta, podemos afirmar que:

a) apenas Sílvio acertou.

b) apenas Cláudio acertou.

c) apenas Célia acertou.

d) apenas os rapazes acertaram.

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

69) Calcule: a) 31 � 29 b) 21 � 19 c) 22 � 18 d) 91 � 89

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________


47

e) 102 � 98 f) 103 � 97 g) 108 � 92 h) 42 � 38

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

________________ ________________ ________________ _______________

i) 28 � 32 j) 55 � 45 k) 51 � 49

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

_______________________ _______________________ _______________________

70) Simplifique a expressão: E = ( )426 24− � ( )4

26 + 24 .

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

71) O valor da expressão 2 24a 4b

3(a + b)(a b)−

−para a = 3,7 e b = 2,9 é:

a) 43

b) 43

− c) 209

d) 209

−

________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

72) Se x = 3 + 1, calcule x2 – 2x + 1.

a) 3 b) 3 c) 4 + 4 3 d) 4

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________


48

CAPÍTULO V – Frações Algébricas

73) Determine o mdc e o mmc dos monômios: a) 12x2, 9x3 b) 8m2n, 20mn3

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

c) 16x2, 20x, 10x3 d) 3x, 6x2y, 9y3

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

e) 12a2x, 16ax3, 20a2x2 f) 6a3b, 9ab2c, 15a2b4c2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

g) 2x5, 5x3, 4x4 h) 60ay2, 24a3y, 12a2y4

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

74) Determine o mdc e o mmc das seguintes expressões algébricas: a) 4x, 2x – 2 b) ab, ab + bc

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________


49

c) 3xy, 5x2 + 5xy d) 2ax, a2 – a, ax – a

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________

e) x + 1, x2 – 1 f) ax + bx, a2 + 2ab + b2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

g) x2 + xy, xy + y2, x2 – y2 h) 5x + 10, 2x + 4, 3x + 6

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________

i) x2 – 25, x2 – 10x + 25, 5x – 25 j) 5x, x2 – 2x, x2 + 2x

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________

k) x2 – a2, 2x + 2a, x2 + ax l) x + 3, x2 – 9, 2x + 6

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________


50

m) 4 – 4a + a2, 4 – a2, 2a2 – a3 n) 2x2 – 2, 2x2 – 4x + 2

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________

o) a2 – 1, 2a + 2, a + 1 p) 2x, x2 + x, x2 – x, x2 – 1

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________

___________________________________ ___________________________________

___________________________________ ___________________________________

75) Simplifique as seguintes frações algébricas:

a) 2a

ab b)

6x12y

c) 2 2

abcb c

d) 2

2

5x y10xy

________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________

e) 3 2

2 2

4a m12a m

f) 3 2

2

6a c9a bc

g) 2

mm + m

h) 2

ac + bcc

________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________

i) 2

2ab2a + 2a

j) 2x + 2yax + ay

k) 2 2

a xa x

−−

l) 2x + x

xy + y

________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________


51

m) 2

1 + 2a4a 1−

n) 2c + 3c

2c + 6 o)

2

2 2

h + bha b−

p) 2

2

x + 6x + 9x 9−

________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________ ________________ ________________ ________________ _______________

q) 2

4

2x + 2x 1−

r) 3

6

x 1x 1

−−

s) bx + cx + by + cy

ax + ay

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

t) 2

2 2

x xyx 2xy + y

−−

u) 2x 16

5x + 20−

v) 3 2

2

x + x + x + 1x 1−

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

w) 24x + 8ax

2x + 4a x)

2 2

3 3

x + xy + yx y−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

76) Determine as seguintes somas algébricas:

a) x 3x

+ 5a 10a

b) 3x 5x x

+ 4y 6y 3y

− c) 2

1 2 + + 1

x x

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


52

d) 3a 2b

+ b a

e) 2

2 5 1 +

a a 2− f)

1 1 1 + +

x y xy

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

g) a 1 a + 1

+ a 2−

h) x + a a x

x a−− i)

x + y x y +

2x 2y−

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

j) 2

1 x +

x + 2 x 4− k)

1x + 1

x 1−

−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

l) a b 2a

+ a + b a b

−−

m) 2

2

x 5y 5y +

x + y xy + y−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________


53

n) 2

2

1 + a 4a 1 a +

1 a 1 a 1 + a−−

− − o)

2

2 2

a aa b a b

−− −

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

77) Determine os seguintes produtos:

a) 2a 2

3x y� b) 2

x y

2a a� c) 2

am xy

x a�

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

d) 3

2

3a x

x 6a� e)

3 2 2

3 2

a b 2x y

10xy a c� f)

2 2x y ab 2

a x by� �

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

g) x + 2 x 1

x 2x

−� h)

x y

x y x + y−� i)

2 2a a b

a b ab−

−�

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


54

j) 2

9x a + 2

a 4 3x−� k) 2 2

ax + x 3x 3y

x y a + 1−

−� l)

3

4 2

5a + 5 x

x + x a + 1�

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

m) 2 2

2 2

a + 2ax + x m n

m n a + x−

−� n)

2

4

x + 1 ab + a 4x + 4

4 x 1 a−� �

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

78) Determine os seguintes quocientes:

a) 3x b

: a x

b) 5a 10

: bc c

c) 2

3a a :

4b b

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

d) 6a 2

: 5bc abc

e) 3

1 2x :

xy y f)

3 2

2 2

8m 4mb :

3ax 3x

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


55

g) 2

2

a a :

x + 1 x 1− h)

x + y 2x + 2y :

x y 2− i) 2 2

am m :

a m a + m−

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

j) 2x 25 2x + 10

: xy x−

k) 23a 3a

: c + 1 2c + 2

l) 2

2

x 2x + 1 x 1 :

x + 2x x− −

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

m) 2 2

2

4 + 4a + a a 4 :

1 b b + 1−

− n)

2 3 2

2

a + a + 1 a + a + a :

a + 1 a 1−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

o)

2

1a1a

p) 2

2

xyaxa

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________

q)

2

aa 22a

a 4

−

−

r) 2

x + yxy

x + xyy

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________


56

79) Determine as seguintes potências:

a) 2

2ax

� ��

b) 23x

y� ��

c) 3

xy2a

� ��

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

d) 32

3

m nx

� ��

e) 4

2

1a c� ��

f) 3

1x + 2

� ��

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

g) 2

a + b2c

� ��

h) 2

3

x ya−� �

� ��

i) 2

x + ax y

� �� −� �

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

j) 1

ab

−� ��

k) 13x

2a

−� ��

l) 12a

m

−� ��

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________

m) 2

2

xy

−� ��

n) 3

2

abm

−� ��

o) 2

ab + c

−� ��

_______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________ _______________________


57

80) Simplifique as seguintes expressões algébricas:

a) 2 2

2 2

(x y) yx 4y− −

− b)

2(x + a)(x a) + aax + bx

−

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

c) x y x

: + 1y x y

� � � �−� � � �

� � � � d)

a b a b + 1

a + b b a−� � � �−� � � �

� � � ��

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

e) 1 1 a b

+ : + 2 + a b b a� � � ��

f) x a x a

1 + : 1x + a x + a

− −� � � �−� � � ��

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________


58

g) 2 2

2 2 2

(x y) y(x + y) x y

− −− −

h) x y x y

1 : + 1x + y x + y

� � � �− −−� � � ��

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________

___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________

81) Sabendo que x ventiladores iguais custam R$ 500,00, pergunta-se: a) Que fração algébrica representa o preço de um deles?

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

b) Ana deu y reais na compra de um deles. Que fração algébrica representa o troco dessa compra?

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

Testes de Revisão 82) Em qual expressão abaixo o número 5 pode ser cancelado sem mudar o valor da fração?

a) x + 5y 5−

b) 5 + x5 + y

c) 5x + 5y

5y d)

5x y5−

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

83) O valor da fração 4 5

4

6 + 66

é:

a) 6 b) 7 c) 36 d) 37

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________


59

84) Simplificando a expressão 2 2 2

2 2 2

a + b + ca b c− − −

, obtemos:

a) 0 b) 1 c) –2 d) –1

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

85) O valor de 4

2

x 1(x 1)(x + 1)

−−

, para x = 1999 é:

a) 2 000 b) 3 000 c) 4 000 d) 5 000

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

86) (Olimpíada Brasileira de Matemática) Se xy = 2 e x2 + y2 = 5, então 2 2

2 2

x y + + 2

y x vale:

a) 254

b) 52

c) 54

d) 12

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________

87) Em uma prova em que deviam ser dados os resultados do 1º membro um aluno desatento apresenta

estes cálculos:

Quantos enganos esse aluno desatento cometeu?

a) 1 b) 2 c) 3 d) 4

__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________


60

Respostas dos exercícios

CAPÍTULO I – Matemática Básica

1a) 10 b) 122 c) 1 d) –3 e) –9

f) 3 g) –10 h) 7 i) –14 j) 1

2a) 10 b) 2 c) 11 d) –15 e) –11 f) –2

3a) (3 000 000 – 1 800 000) : 3

b) 26 + 3 � 26

c) (385 : 5) – 32 � 5 ou 385

32 55

− �

d) (960 – 336) : 8 ou 960 336

8−

e) 2 � 27 + 2 � 26

f) 250 + 4 � 140

g) 4 + 2 � 12

h) (814 – 94) : 5 ou 814 94

5−

CAPÍTULO II – Cálculo Algébrico

Parte I – Monômios

4a) –2a b) 2xy c) –6ac d) –3am e) a2 f) 6xy2 g) 9

bc5

h) 21x

10 ou

2x10

i) 5

mn4

− j) 4x k) –4y2 l) ab m) 0 n) 4xy

o) 3n3 p) –5am q) 41a

6 ou

4a6

r) 2

bc5

−

5a) 3y3 – 3y b) a – 2ab + 4b c) 9x2 + 2x – 1 d) 3mn + 2m + n

e) a2 – ab + 9b2 f) 3x – 2y + 4 g) 3 4

a b2 3

− h) 4x2 + 5

x8

ou 28x + 5x

8

6) P = 3x

7a) x2 + 5y b) x2 + y2 c) (x + y)2 d) (a + b)(a – b) e) 2a + 2h

f) a3 + b3 g) (a + b)3 h) x2 – y2 i) 2x

3 j) x – 5

8) A = 100x2 9a) 0,40x + 3y b) 0,30x + 2y c) 0,10x + y 10) 5,87 � 157

11a) –13 b) 15 c) 13

d) 1 e) 144 f) 16


61

Parte II – Polinômios 12a) 3x – 2y b) 4a2 – a – 4 c) 11x3 – 4x2 – 7x + 6 d) a – 11b + 5c e) –y2

f) 2ab + bc g) 2a – 1 h) 3a – d i) 3x – 10y + 16 j) 3x – y

13a) –27y2 b) 10x2y c) 24bc d) –x3y5 e) 10x4y3

f) –12m3nx3 g) 5 23x y

10− h) 3 25

p q4

i) 10x6 j) –18a3x2

k) –60a2b4 l) –m2x5y3 m) –x4y4 n) 4 26h x y

5− o) –30k2x4 p) 215

abn px2

−

14a) 4y3 b) 5x2 c) –3 d) x e) –1

f) 2y g) –a3b h) 4m i) 1

ab2

− j) 5

x3

k) 21n

2− l)

1a

8 m) –6a2b4c n) ab2c

15a) 16x6y2 b) –m3x6 c) 32a5c15 d) 81b8c4 e) h12m6 f) 121y

8−

g) 2 24m n

25 h) x10y25 i) 1024a30 j) 1 k) 181

a64

l) 0,25x8y14

16a) –2m2 + 7m + 2 b) 2m2 – 7m – 2 c) 5m2 + 6m

d) 7m2 – m – 2 e) –5m2 – 6m f) –7m2 + m + 2

17a) 5 3

a c4 2

− b) 1

10 c) 211 1

m 8m + 2 4

− d) 2 5x 2x +

2−

18a) 3 + 4x b) –3x – 14 c) 1

+ 5x2

19a) 5ax3 + 5bx2 + 5cx

b) –15x2y2 – 18xy3 + 21x2y3

c) m3n – m2n2 + nm3

d) 6ax – 8bx + 10cx

e) 7a2x – 21abx + 35acx

f) 6x3 – 4x2 + 2x

g) 12x3 + 9x2 – 15

x2


62

20a) x3 – 12x2 + 47x – 60 b) x3 – 5x2 – 6x + 30 c) x4 – 7x3 + 6x2 + 42x – 72

d) x4 – 6x3 + x2 + 36x – 42 e) x4 – 8x3 + 18x2 – 5x – 12 f) 2x4 – 20x3 + 67x2 – 79x + 12

21) 2 ab2a +

2

22a) 4x + 3 b) –3x + 2 c) x3 + 5x2 + x – 4

d) 4a3 – 3a2 + 5 e) 2m3 – 7m2 + 3m f) –4x + 2 + 3a10

23a) x + 2 b) 2x – 3; resto 2 c) 3x – 5

d) 2x2 – 3x + 3; resto 1 e) –5x2 + 3x – 7 f) x – 5

g) x2 + 2x – 1 h) –2x + 1 i) 3x + 9; resto –6x – 7

j) x2 + x + 3; resto x + 4 k) 4x2 + 6x+2; resto –3x – 6 l) x2 + 4x + 16

24) 9x2 + 21x 25) 0,70x + 0,12y

26) A(1) = A(3) = A(3) = 2x

2 A(2) = A(4) =

2x4

A(6) = A(7) = x2

27a) 18 28) 27

CAPÍTULO III – Produtos Notáveis

36a) x4 + 2x2y3 + y6 b) 24a2x4 + 60a4x2 + 36a6 c) 16a6x2 + 16a3bxy + 4b2y2

d) a4 + 10a3m + 25a2m2 e) x12 + 6x8 + 9x4 f) 2 225x 20 4y

+ xy + 9 2 9

g) 8 4 19x + 2x +

9 h)

43 29a 3

+ a + a16 2

i) 6 7 8x x x

+ + 9 6 16

j) 2 4 4 2

3 3a b 2 a b + a b +

9 9 9 k) a6 – 8a4 + 16a2 l) 9x4 – 30x3y + 25x2y2

m) 24a4 – 40a2b2 + 16b4 n) x6 – 2x3y3 + y6 o) 4a2x2 – 16a3x4 + 16a4x4

p) 6 38 16x x +

3 9− q) 4 2 5 6 24 2 1

a b a bx + a x9 3 4

− r) 4 2 2 2

4m x m xy 9 + y

16 5 25−

s) 6 2 3 2 24 2 25a x a bxy + b y

9 3 4− t)

126x 5 25

+ x + 4 4 16


63

37a) a4 – 1 b) a6b4 – c2 c) x6 – y6 d) 25a6b2 – 4x2 y4

e) x4y8 – 25x2 f) 4x 9

25 16− g)

49m 14 9

− h) 2

6 ba

25−

i) 425x

369

− j) 8x

369

−

38a) a6 + 3a4b2 + 3a2b4 + b6 b) 8a3x3 + 36a4x2 + 54a5x + 27a6

c) x9 + 3x7 + 3x6 + x3 d) 3

227x 9 1 1+ x + x+

8 4 2 27

e) 6 4 2 2 3a a b a b b

+ + +64 16 12 27

f) 3 6 4 5 2 4 3 3a x a bx ab x b x

+ + +8 4 6 27

g) x6 – 3x4y + 3x2y6 – y9 h) 27a6 – 27a4b3x + 9a2b6x2 – b9x3

i) m6 – 9m4xy2 + 27m2x2y4 – 27x6y6 j) 6 4 2x x 9x 27

+ 27 4 16 64

− −

k) 9

3 61 3 3 aa + a

8 16 32 64− − l)

6 4 2 2 3x x y x y y +

8 8 6 64− −

39a) x2 + y2 + 9 + 2xy + 6y + 6x b) x4 + y2 + 1 + 2x2 y + 2x2 + 2y

c) 4x2 + y2 + 1 – 4xy – 4x + 2y d) x2 + 16y2 + 9 – 8xy + 6x – 4y

e) m10 – 2m5 + 1 f) 64x2y2 – 4xy + 1

16 g) x3 + 15x2 + 75x + 125

h) x3 – 3x2c + 3xc2 – c3 i) 25x10 + 20x6 + 4x2 j) 4 125m

4−

k) 25m6x2 – 49n6z4 l) 9a2 + 12ay + 4y2 m) 4 2 3 69m 15m n + 25n

4−

n) 4 216 4a b

25 81− o) 3 3 3 2 3 31

x y x y + 9x y 27x27

− − p) 4x2 + 2x2y + 41y

4

40a) x2 + 10x + 24 b) a2 – 8a + 15 c) y2 – 4y – 21 d) m2 – 4m – 96

e) x2 – 4x – 45 f) x2 + 18x + 45 g) b2 + 16b – 80 h) k2 – 21k + 104

i) x2 + 5x – 66 j) u2 + 11u – 60 k) 2 5 3x + x +

4 8 l) 2 16

x + x 115

−

m) 2 25x x 6

3− − n) 2 19 3

x x + 10 5

− o) 2 1 14x + x

3 3− p) 2 3 21

x x4 16

− −

41) 29 42) 83 43) 425 44a) 2499 b) 39 996 c) 4851 d) 399


64

45) 404 46) 8 47) 3x2 + 2x – 4 48) 7

50) 16; 22

44x + 8 +

x 51) x2 – 7x + 70 52) 198

125− 53) alternativa c

CAPÍTULO IV – Fatoração 54a) a(a2 – x) b) 5ax(1 – a2x) c) p(7p + 1)

d) 15a2(1 – 15a2) e) 5(3 + 5x2) f) 2x3(3 + x + 2x2)

g) 2xy2(xy – 3x + y) h) 3a2(a2 – 2a2 – 3a3) i) 5x3(x2 – 2a2 – 3a3)

j) 3x(2x2 – 3xy + 4y2) k) 2 3 21 1 1

a b a bc + a b + 12 6 3

� �−� ��

l) 3 5 5 21 1 1 1m n a m mn + n

4 5 5 2� �−� ��

m) 2 3 33 2 3 3h k x y + k

5 3 4 2� �−� ��

n) 3 4 4 3 21 2 1a b a b + 2a

3 5 4� �−� ��

o) 2 4 6 4 56 1 2 3y z z + y z

5 7 9 5� �−� ��

55a) (a + b)(a + c) b) (2 + c)(x + c) c) (5 + b)(a + b) d) (x – y)(m – n)

e) (a – c)(a + x) f) (3a – b)(x – y) g) (2x + y)(3x – a) h) (ax – 3by)(x – y)

i) (x – 3)(x – y) j) (x + 1)(x3 + 2) k) 1 1

a + 3 5� ��

(a2 + 1) l) (x + my) 1

x y4

� �−� ��

m) 2 1 m 2m

3 2 3� �� − −� ��

n) (3a + 2)(2m + 5n) o) (a3 – b)(a2 + 1) p) (4m + 5n)(9a + 1)

56a) (r – x)(r + x) b) (a – 2)(a + 2)

c) (m – 3)(m + 3) d) (b + 4)(b – 4)

e) (5 + y)(5 – y) f) (2 + x)(2 – x)(4 + x2)

g) (a2 – 10)(a2 + 10) h) (np – 1)(np + 1)

i) (x – y)(x + y)(x2 + y2) j) (4x + 5y)(4x – 5y)

k) (x3 – 12a2)(x3 + 12a2) l) (x2 + ab)(x2 – ab)

m) (3a2 + 4n)(3a2 – 4n) n)

1 1y + y

2 2� �� −� ��

o) a b a b

+ 3 4 3 4� �� −� ��

p) x y x y

+ a b a b

� �� −� ��

q) mr xp mr xp

+ 4 5 4 5

� �� −� ��

r) –4ab

s) 4a t) –5(2a + 1)


65

57a) (2x + 1)2 b) (x + 5)2 c) (2x2 + y3)2

d) (4 – x)2 e) (6mn2 – 2x3)2 f) (5x2y – 2p3)2

g) (2x5 + 3y2m)2 h) (2x – 1)2 i) (x – a)2

j) 2

33 bx y

5 4� �−� ��

k) 2

41 3m

4 2� �−� ��

l) 2

3 4 21 1a + b c

4 2� ��

m) 2

3 12x + y

3� ��

n) 2

2 3 216m y + xy

2� ��

o) (x – y4)2

p) não é fatorável q) (2x – 1)2 r) não é fatorável

s) não é fatorável t) não é fatorável u) 2

1x + 1

3� ��

58a) (x + 3)(x + 4) b) (x + 2)(x + 5) c) (x – 1)(x – 6) d) (x – 2)(x – 4)

e) (x – 2)(x – 7) f) (x – 3)(x + 4) g) (x – 3)(x – 6) h) (x – 1)(x – 8)

i) (x – 4)(x + 3) j) (x – 2)(x + 6) k) (x – 1)(x + 8) l) (x + 3)(x – 5)

m) (x – 2)(x + 3) n) (x – 1)(x – 4) o) (y + 1)(y – 12) p) (m – 1)(m – 12)

q) (t + 2)(t + 6) r) (a – 4)(a + 2) s) (k + 5)(k + 8) t) (z + 1)(z – 8)

59a) x(x + 5) b) (2x – 3)2 c) (x – 2)(x2 + 4) d) (2x + 3)(2x – 3)

e) a3(a3 – 5a2 + 6) f) (x – 1)(a + b) g) (8y + 5)2 h) a2b2(a + b)

i) (m3 + 1)(m3 – 1) j) (2ax – b)2 k) 6a(2ab + 3) l) (x – y)(x2 + y)

m) (x + 4)(x – 2) n) abc(a + b + c) o) (5x + 7)2 p) (1 + a + b)(1 – a – b)

q) (x2 + 1)(x4 + 1) r) 5a2m(3a – 4) s) 1 1

m + 5n m 5n2 2

� �� −� ��

t) (9y + 1)2

u) (x +1)(x + 2) v) (m – 1)(m + 4) w) (m – 3)(m + 1)

x) (x – 2)(x + 5) y) (m – 2)(m + 1) z) (x – 5)(x – 6)

60a) 10(a + 1)(a – 1) b) x(x – 5)2 c) 2(m + 2)(m – 2) d) a(y + 2)2

e) (h2 + m2)(h + m)(h – m) f) y(x + 6)(x – 6) g) (a + c)(b + 1)(b – 1) h) x2 (x + 1)2

i) (9 + k2)(3 + k)(3 – k) j) xy(x – 4y)2 k) y(x – 4)2 l) 4(z – 15)2

m) 5(x + 2)(x – 2) n) z(2x + 5)(2x – 5) o) 12a(x2 – a)2 p) 4x(x – 1)2

q) (a + 1)(a + 2)(a – 2) r) (x – y)(x2 + xy + y2)(x + y)(x – xy + y2)

s) 12a(b – c) t) –x(x2 + x – 3)(x2 – x + 3)

61) 2006 62a) 99 b) 3999 c) 196 d) 4001 e) 1


66

64) Não, não pode ser um número par. 65) 500 66) 540

67a) (x3 + y2)(x6 – x3y2 + y4) b) (m2 + 2n4)(m4 – 2m2n4 + 4n8)

c) (y4 – 3)(y8 + 3y4 + 9) d) (2a – 3b)(4a2 + 6ab + 9b2)

e) (4 + a2)(16 – 4a2 + a4) f) (5x2 – 1)(25x4 + 5x2 + 1)

g) (x2y – a3b4)(x4y2 + x2ya3b4 + a6b8) h) 3 6 3x 4 x 2x 16

+ + 2 2 4 3 9

� �� −� ��

� ��

i) 5 2 10 5am a m am

1 + + 15 25 5

� �� −� ��

� �� j) 4 8 3 4 62 6 4 4 36

x + a x a x + a3 5 9 5 25

� �� −� ��

k) 2

2 4 21 3a b 13a b 9a + + b

2 2 4� �� − � ��

� ��

68) alternativa B 69a) 899 b) 399 c) 396 d) 8099 e) 9996

f) 9991 g) 9936 h) 1596 i) 896 j) 2475 k) 2499

70) 16 71) alternativa A 72) alternativa A

CAPÍTULO V – Frações Algébricas

73a) mdc = 3x2 / mmc = 36x3 b) mdc = 4mn / mmc = 40m2n3

c) mdc = 2x / mmc = 80x3 d) mdc = 3 / mmc = 18x2y3

e) mdc = 4ax / mmc = 240a2x3 f) mdc = 3ab / mmc = 90a3b4c2

g) mdc = x3 / mmc = 20x5 h) mdc = 12ay / mmc = 120a3y4

74a) mdc = 2 / mmc = 4x(x – 1)

b) mdc = b/ mmc = ab(a + c)

c) mdc = x/ mmc = 15xy(x + y)

d) mdc = a/ mmc = 2ax(a – 1)(a + 1)

e) mdc = (x + 1) / mmc = (x + 1)(x – 1)

f) mdc = (a + b) / mmc = x(a + b)2

g) mdc = (x + y) / mmc = xy(x + y)(x – y)

h) mdc = (x + 2) / mmc = 30(x + 2)

i) mdc = (x – 5) / mmc = 5(x – 5)2(x + 5)

j) mdc = x / mmc = 5x(x – 2)(x + 2)

k) mdc = (x + a) / mmc = 2x(x + a)(x – a)

l) mdc = (x + 3) / mmc = 2(x + 3)(x – 3)

m) mdc = (2 – a) / mmc = a2(2 – a)(2 + a)

n) mdc = 2(x – 1) / mmc = 2(x – 1)2(x + 1)

o) mdc = (a + 1) / mmc = 2(a + 1)(a – 1)

p) mdc = 1 / mmc = 2x(x + 1)(x – 1)

67

75a) ab

b) x2y

c) abc

d) x2y

e) a3

f) 2ac3b

g) 1

m + 1

h) a + b

c

i) b

a + 1

j) 2a

k) 1

a + x

l) xy

m) 1

2a 1−

n) c2

o) h

a b−

p) x + 3x 3−

q) 2

2x 1−

r) 3

1x + 1

s) b + c

a

t) x

x y−

u) x y

5−

v) 2x + 1

x 1−

w) 2x

x) 1

x y−

76a) x2a

b) 5x4y

c) 2

2

x + 2x + 1x

d) 2 23a + 2bab

e) 2

2

4a + 10 a2a

−

f) y + x + 1

xy

g) 2a + 3a 2

2a−

h) 2 2a + yax

i) 2 2x + y2xy

j) 2x 2

(x + 2)(x 2)−

−

k) 2x 2

x 1−−

l) 2 23a + b

(a + b)(a b)−

m) x

x + y

n) 4a

1 a−

o) ab

(a + b)(a b)−

77a) 4a

3xy b) 3

xy2a

c) mya

d) 2a

2x e)

2

2

ab x5cy

f) 2xy g) 2

2

x + x 22x

− h) 2 2

xyx y−

i) a + b

3 j)

3a 2−

k) 3x

x + y l) 2

5xx + 1

m) a + xm + n

n) b + 1x 1−

78a) 23x

ab b)

a2b

c) 3b4

d) 23a

5 e)

2

2

y2x

f) 2mab

g) x 1

a−

h) 1

x y− i)

aa m−

j) x 52y−

k) 2a

l) x 1x + 2

− m)

2 + a(1 b)(a 2)− −

n) a 1

a−

o) a

68

p) ayx

q) a + 2

2 r) 2

1x

79a) 2

2

4ax

b) 6

2

xy

c) 3 3

3

x y8a

d) 6 3

9

m nx

e) 8 4

1a c

f) 2

1x + 4x + 4

g) 2 2

2

a + 2ab + c4c

h) 2 2

6

x 2xy + yx

− i)

2 2

2 2

x + 2ax + ax 2xy + y−

j) ba

k) 3

2ax

l) 2

ma

m) 4

2

yx

n) 6

3 3

ma b

o) 2 2

2

b + 2bc + ca

80a) a

x + 2y b)

xa + b

c) x y

x−

d) 2(a b)

b−

e) 1

a + b f)

xa

g) x 2y

2y−

h) yx

−

81a) 500

x b)

xy 500x−

82) alternativa C 83) alternativa B 84) alternativa D 85) alternativa A 86) alternativa A 87) alternativa D

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