Polynomials – Review Booklet Multiple...

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Polynomials – Review Booklet

Multiple Choice 1. Determine the greatest common factor of 24x3y4 – 15xy2 – 36x2y3 :

A. xy B. 4xy2 C. 3x2y2 D. 3xy2

2. What is the greatest common factor of 24x4y3z2 and 36x2y4?

A. x2y3

B. 12 x4y4

C. 12 xy

D. 12 x2y3

E. 36 x2y3

3. To completely factor 2 x6y – 16 x5y – 4 x4y, what is the greatest common factor that will be taken out first?

A. 4 x3y2

B. 2 x4y

C. 2 x6y

D. 2 x5y

E. 2 x4y2

4. Completely factor this expression: 18xy3 + 48x3y2 − 36xy2

A. 6xy3 3+ 8x − 6y

B. 3 3 2 26(3 8 6 )xy x y xy+ −

C. 2 26 (3 8 6 )xy y x y y+ −

D. 2 26 (3 8 6)xy y x+ −

5. Factor completely: 4 336 24x x−

A. 2 24 (9 6 )x x x− B. 3 26 (6 4 )x x x− C. 32 (18 12 )x x x− D. 312 (3 2)x x − E. 2 2 2(9 4 )(4 6 )x x x x+ −

6. When 6 4 2 2 212 3 15m n m n m n− + is completely factored, one of the factors will be _____.

A. 44 5m mn n− + B. 23m n C. 2 24 5m m n n− + D. 2 23m n E. 2mn

7. The polynomials 24 8 5x x+ − and 26 3x x− have in common a factor of

A. 4 1x +

B. 4 1x −

C. 2 1x +

D. 2 1x −

8. In fully factored form 18xy3 + 48x3y2 – 36xy2 is

A. 6xy2(3y + 8x2 – 6)

B. 6xy(3y2 + 8x2y – 6y)

C. 6(3xy + 8x3y2 – 6xy2)

D. 9(2xy3 + 6x3y2 – 4xy2)

9. A factor of the algebraic expression: x2 – x – 6 is:

A. (x – 2)

B. (x – 3)

C. (x + 3)

D. (x – 6)

10. Factor: x2 + 12x + 36

A. (x + 4)(x + 8)

B. (x + 4)(x + 9)

C. (x + 6)(x + 6)

D. (x – 6)(x – 6)

E. (x + 6)(x – 6)

11. Factor x2 – 7x + 12

A. (x + 4)(x – 3)

B. (x – 2)(x – 6)

C. (x – 12)(x – 1)

D. (x + 7)(x + 5)

E. (x – 4)(x – 3)

12. For which of the following is (a – 3) not a factor?

A. 2 3a a− B. 2 6 9a a− + C. 22 4 6a a− − D. 3 29a a−

13. Factor: 5x2 – 14x + 8

A. (5x + 2)(x + 4)

B. (5x + 4)(x + 2)

C. (5x – 2)(x – 4)

D. (5x – 4)(x – 2)

E. (5x + 8)(x + 1)

14. Factor: 8x2 – 18x + 9

A. (8x + 9)(x + 1)

B. (8x + 1)(x + 9)

C. (4x – 3)(2x – 3)

D. (4x + 3)(2x + 3)

E. (4x + 1)(x + 8)

15. Factor 9x2 + 24x + 16

A. (3x + 4)(3x + 4)

B. (4x + 3)(4x + 3)

C. (x + 16)(9x + 1)

D. (3x – 4)(3x – 4)

E. (3x + 4)(3x – 4)

16. Factor: 4x2 – 20x + 25

A. (2x + 5)(2x + 5)

B. (2x + 5)(2x – 5)

C. (4x – 5)(x – 5)

D. (x – 25)(4x – 1)

E. (2x – 5)(2x – 5)

17. Factor completely: 6x2 + 24x + 24

A. (6x + 12)(x + 2)

B. (2x + 4)(3x + 6)

C. 6(x2 + 4x + 4)

D. 6(x + 2)2

E. 2(3x + 2)(x + 6)

18. Factor completely: 4x2 + 24xy + 36y2

A. (4x + 12y)(x + 3y)

B. 4(x + 3y)2

C. (2x + 6y)(2x + 6y)

D. 4(x2 + 6xy + 9y2)

E. 2(2x + 3y)(x + 6y)

19. When 4x2 – 9y2 is completely factored, one of the factors will be _____.

A. x + 3y

B. x – 3y

C. 4x + 3y

D. 2x – 3y

E. 4x – 3y

20. When 9x2 – 42xy + 49y2 is completely factored, one of the factors will be ______.

A. x + 7y

B. 9x – 7y

C. 3x – 7y

D. 3x + 7y

E. 3x – y

21. An equivalent expression to the algebraic expression 9x2 – 4y2 is:

A. 3x – 2y

B. 6(3x2 – 2y2)

C. (3x + 2y)(3x – 2y)

D. (2y + 3x)(2y – 3x)

22. Which expression is a perfect square when factored?

A. x2 – 4x + 4

B. a2 – 4a – 4

C. b2 – 9b + 9

D. m2 – 9m – 9

23. Factor 3x2 – 27

A. 3(x + 3)(x – 3)

B. 3(x – 3)(x – 3)

C. (3x + 3)(x – 9)

D. (x + 3)(3x – 9)

E. (3x – 3)(x – 9)

24. Factor completely: 6x2 – 54

A. 6(x – 3)(x – 3)

B. (6x + 6)(x – 9)

C. (2x + 6)(3x – 9)

D. 6(x + 3)(x – 3)

E. (3x – 3)(2x – 18)

25. Simplify: ( ) ( )2 25 7 3 2 5 9 10a a a a− + − + − −

A. −15a2 + 45a − 33

B. −25a2 + 45a − 33

C. −25a2 − 25a − 3

D. −3− 25a2 − 25a4

26. Simplify: ( ) ( ) ( )2 2 2 2 2 2 29 8 7 6 5 4 3x xy x y xy x y x y xy+ + − − + −

A. 9x2 + 5xy + 7x2y − 6xy2 + 9x2y2 B. 9x2 +11xy + 7x2y − 6xy2 − 9x2y2 C. 9x2 + 5xy + 7x2y − 6xy2 − 9x2y2 D. 9x2 +11xy + 7x2y − 6xy2 + 9x2y2

27. The area of a regular soccer field can be represented by the expression 2 9 112x x− − . If a second soccer field differs in area by a factor of x +1, then what is the area of the second soccer field?

A. 2 11x −

B. x −11

C. x + 9

D. 2 9x +

28. An equivalent expression to the algebraic expression: (x – 2)(3x + 4) is:

A. (x + 2)(3x – 4)

B. 4x + 2

C. 3x2 – 2x + 2

D. 3x2 – 2x – 8

29. Given (2x + 5)(x – 3) = Ax2 + Bx + C, what is the value of B?

A. –x

B. –2

C. –1

D. 1

E. 2

30. Given (-2x + 5)(3x – 4) = Ax2 + Bx + C, what is the value of A + B?

A. –29

B. –17

C. 17

D. 19

E. 29

31. Given (–3x – 7)( –5x + 2) = Ax2 + Bx + C, what is the value of A + B?

A. –44

B. –14

C. 14

D. 17

E. 44

32. Expand and simplify: (2x – 5y)(3x + 2y)

A. 2 26 11 10x xy y+ −

B. 2 26 11 10x xy y− −

C. 2 26 3 10x xy y+ −

D. 2 26 4 10x xy y+ −

E. 2 26 10x y−

33. Expand and simplify: (2x + 5)2 – 3(x + 7)

A. 24 17 4x x+ +

B. 24 17 46x x+ +

C. 24 7 4x x+ +

D. 24 13 4x x+ +

E. 24 7 4x x+ −

34. Expand and simplify : (2x – 5)2(x + 3)

A. 3 24 8 5 75x x x− − +

B. 3 24 12 25 75x x x+ + +

C. 3 24 8 35 75x x x− + +

D. 3 24 32 35 75x x x− − +

E. 3 24 8 35 75x x x− − +

35. Expand and simplify: (3x – 2)3

A. 3 227 54 36 8x x x− − −

B. 3 227 54 36 8x x x− + −

C. 3 227 54 36 8x x x+ + −

D. 3 227 54 36 8x x x− + +

E. 3 227 54 36 8x x x+ + +

36. Multiply: (x – 3)(x2 + x + 3)

A. 3 22 9x x+ −

B. 3 22 9x x− −

C. 3 22 2 9x x x+ − −

D. 3 22 9x x− +

E. 3 22 2 9x x x− + −

37. Multiply: (2x2 + 3x – 4)(2x + 5)

A. 3 24 16 7 20x x x+ + −

B. 3 24 16 7 20x x x+ + +

C. 3 24 16 7 20x x x+ − −

D. 3 24 16 23 20x x x+ + −

E. 3 24 16 23 20x x x+ − −

38. Multiply: (3x – 2)(9x2 + 6x + 4)

A. 3 227 12 18 8x x x+ − −

B. 3 227 36 54 8x x x− + −

C. 327 8x −

D. 3 227 36 54 8x x x+ + −

E. 3 227 36 54 8x x x− − −

39. Multiply: (2x2 – 3x + 4)(x2 +5x – 2)

A. 4 3 22 7 15 26 8x x x x+ − + −

B. 4 3 22 7 15 26 8x x x x− − + −

C. 4 3 22 7 15 26 8x x x x+ + + −

D. 4 3 22 7 15 14 8x x x x+ − + −

E. 4 3 22 7 19 26 8x x x x+ − + −

40. Multiply: (4x – 3)(3x3 + 5x – 3)

A. 4 3 212 9 20 27 9x x x x+ − + +

B. 4 3 212 9 20 27 9x x x x− + − +

C. 4 3 212 9 20 27 9x x x x− − − −

D. 4 3 212 9 20 27 9x x x x+ + + −

E. 4 3 212 9 20 27 9x x x x− − + +

41. Given (x2 – x + 2)(x – 1) = Ax3 + Bx2 + Cx + D, what is the value of A + D?

A. – 2

B. – 1

C. 0

D. 2

E. 3

42. If (8x2 + 4x + 7)(3x2 + 5x + 6) = 24x4 + 52x3 + Bx2 + 59x + 42, then the value of B is

____.

A. 24

B. 41

C. 69

D. 89

E. 94

43. Simplify: ( )( ) ( )( )4 2 2 4x x x x− + − − +

A. –4x

B. 4x

C. 2x2 −16

D. 2x2 +16

44. Expand: 2(9 4)x −

A. 81x2 +16

B. 81x2 − 72x +16

C. 81x2 + 72x −16

D. 81x2 + 72x +16

45. (10y + 4z)2 equals

A. 100y2 + 40yz + 16z2

B. 100y2 + 80yz + 16z2

C. 100y2 + 16z2

D. 196yz2

Use the diagram below to answer this question

46. The area of the rectangle shown above is equal to:

A. 2x2 + 10x – 5

B. 6x + 8

C. 2x2 + 9x – 5

D. 3x + 4

Use the diagram below to answer this question:

47. What is the area of the shaded region in the diagram above?

A. x2 – 2x

B. 6x2 + 11x + 3

C. 2x2 + 3x – 5

D. 5x2 + 15x + 5

48. What is the value of the expression 3x2 – 4x when x = 5?

A. 10

B. 50

C. 55

D. 95

E. 355

II. Numerical Response Place the answers in the numerical response boxes provided.

Start on the left side.

1. When x2 + x − 56 is factored in the form ( )( )x x+ −a b , then the value of

a is ________________( Record your answer in the first column) b is ________________( Record you answer in the second column)

2. When 24 21 18 is factored in the form of (4 )( + )x x x x+ − − b d , then the value of

b is ________________( Record your answer in the 1st column) d is ________________( Record you answer in the 2nd column)

3. When (3x – 2) and (2x + 5) are multiplied, the product is 6x2 + bx −10 where b is __________.

4. Determine the value of the constant term when the following multiplication is performed : ( )( )94342 2 −−− xxx

III. Written Response Show all work. Marks are awarded as indicated.

1. Completely factor each of the following expressions. Show all work as necessary.

• 4x3y4 – 12x2y7 – 16xy2 = ____________________________________________

• 2 13 30x x− − = ____________________________________________

• 4 81x − = ____________________________________________

• 4 28 – 2 – 3x x = ____________________________________________

• x3 + 2x2 – 3x = ____________________________________________

• 3x2 – 10x + 8 = ____________________________________________

• 24 36x − = ____________________________________________

• 26 35x x− − = ____________________________________________

• 23 30 72a b ab b− + = ____________________________________________

• 2 2242 2x y− = ____________________________________________

• 26 2 4x y xy y+ − = ____________________________________________

• 214214 2 +− yy = ____________________________________________

• 6 5 42x x− − = ____________________________________________

2. Simplify each of the following expressions

• 8 10 3 2 14xy x z x yx z+ − − + −

• 2 2(5 3 4) (3 1)x x x x− + + − −

3. Expand and simplify each of the following questions

• 2 2 2 2(2 2 ) ( 2 )m mn n m mn n+ − − − −

• 2 24(2 3) 2(6 5 7)a a a a− + − + −

• (2 1)(3 5)x x+ −

• 2( 3 )(2 5 )x y x y+ −

• 2(7 3)x −

• 22(3 1)m +

• 2(2 1)(3 4 5)x x x+ + −

• ( 6)( 4) ( 1)(5 2)x x x x+ − + + −

• 3(2 7)( 6) 5( 3)( 3)x x x x+ − − + −

• ( )( ) ( )( )312323 +−−−+ mmmm

• (3x3 +2x – 5)(x2 + x – 3)

• (4x3 – 3x2 + 2x – 1)(4x3 + 3x2 + 2x + 1)

4. Evaluate 3 25 2x y+ if x = 3 and y = −2

Use the following information to answer this question.

5. a. Determine the perimeter of this complex shape algebraically and express your answer in its simplest form.

b. Determine the area of this complex shape algebraically and express your answer in its simplest form.

The following complex shape is missing the measurements of two sides. Determine the algebraic expression for each of the missing sides. Write your answers on the blanks provided.

2x + 3

3x – 1

2x + 1

x + 2

Use the following information to answer this question.

6. a. Determine algebraically an expression which represents the new width of the larger rectangle.

b. Determine algebraically, the simplified expression in expanded form for the area of the larger rectangle.

A rectangle is changed into a larger rectangle by increasing the width by 1 unit as shown in the diagram below. The length remains the same.

2x

x2 + 4x – 5 x2 + 4x – 5

7. Annie Pelletier won a bronze medal for Canada in women’s springboard diving at the

Summer Olympics in Atlanta. She dove from a springboard with dimensions that can be represented by the binomials 27 −x and 10−x .

a. Multiply the binomials.

b. If x represents 70 cm, what was the area of the board, in square centimetres.

8. A box has dimensions of 2 5x + by 3 3x − by 4 2x + . Find the volume of the box.

9. a. Find the perimeter of the interior rectangle.

b. Find the area of the shaded region.

3x - 2

2x + 7 4x + 5

x + 1