Week 2: POLYNOMIALSSAT Prep

I. POLYNOMIALS

A.) Vocabulary

Monomials – Any number or variable or product of number(s) and variable(s)

 Ex. Evaluate when a = -4 and b = 0.5.

 

23a b

23 4 0.5 3 16 0.5 24

B.) Simplifying Polynomials

!!** ADD OR SUBTRACT LIKE TERMS ONLY **!! 

Like terms – same variable(s) and same exponent(s). 

Ex. Simplify  

 

2 23 4 2 5 . x x x x

2 23 2 4 5 x x x x 25 9x x

Binomials – 2 monos separated by +/- 

Trinomials – 3 monos separated by +/-

2 3x

22 3 1 x x

Multiplying Monomials - Mult. Coeffecients and add exponents of like bases

 Ex. Simplify the following:

  

2 3 2(3 )( 2 )xy z x y

2 2 33 2 x x y y z 3 3 36 x y z

Dividing Monomials- Div. coeff. And subtract exponents of like bases 

Ex. Simplify the following:

  

4 2 3

2

(6 )

( 2 )

x y z

x z

4 2 3

2

6

2 1

x y z

x z

4 2 2 3 13

1 1 1 1

x y z

2 2 23 x y z

C.) Factoring and Expanding

FOIL – First Outer Inner Last

 Ex . Expand the following:

    

3 3 7 x x

FIRST:

3x x

23 2 21 x x

OUTER: INNER: LAST:

7x 3 3x 3 7

23x 7 x 9 x 21

Three important binomial products

2 2

2 2 2

2 2 2

( )( )

( ) 2

( ) 2

x y x y x y

x y x xy y

x y x xy y

 Ex. If (a – b) = 17.5 and (a + b) = 10, what does a2 – b2 =?   

 Ex. If x2 + y2 = 36 and (x + y) 2 = 64 what is xy?   

2 2a b a b a b

175 2 2 17.5 10a b

2 2 22x y x xy y

2 264 2x xy y

28 2xy

64 2 36xy

14xy

D.) MORE FACTORING GCF, Common Monomials, and Product/sum table 

Ex. Find all real solutions of    

 

2 6 0x x

FACTORS (-6) SUM (-1)

1,6

1, 62,3

2, 3

5

5

11

2 3 0 x x

2 0 or 3 0 x x

2 or 3 x x

Ex. Find an equivalent expression for     

  Ex. Find the sum of reciprocals of  

 

2

2

3 4

4 4

x

x x

2

2

3 12.

4 4

x

x x

2 2

1 1 and .

x y

3 2 2

2 2

x x

x x

3 2

2

x

x

2 2+ x y

A.) Single Equations

Ex. Solve the following for x:     

II. EQUATIONS and INEQUALITIES

13( 2) 2( 1) 1

2x x x

13 6 2 2 1

2 x x x

12 3 6 2 3

2 x x x

6 12 4 6 x x x

7 12 4 6 x x

3 18x

6x

 Ex. If a = b(c + d), solve for d in terms of a, b, and c.  

 

 Ex. If , then x = ?  

3 1 5 x

3 6x

a bc bd

a bc bd

a bcd

b

or a bc a

d d cb b

2x

2 2

2x

4x

Ex. If 2x – 5 = 98, then 2x + 5 = ?   

  Ex. For what value of x is ?    10

2 5 2 5 10 x x

4 3 10

5

x x

2 5 98 10 x 108

3 6

5x

3 30x

Ex. If , solve for x.   

  Ex. If , then x = ?   

25 3 y x

2 4 125 x

11x

23 5 y x

23

5

yx 3

5

yx

2 121x

2 121x

Ex. Find the largest value of x that satisfies .    

  Ex. If , what is w?   

22 3 0 x x

2 3 0 x x

0 or 2 3 0 x x 3

2x

3 14 8 w w

3 12 32 2

w w

2 3 3 12 2 w w

2 3 3 1 w w

2 6 3 3 w w

9w

B.) Systems of Equations/Inequalities 

Use appropriate method Substitution, elimination, graphing, matrices

 Ex. Solve for x and y if x + y = 10 and x – y = 2.    

10

2

x y

x y

2 12x

6x

6 10 y

4y

6,4

Ex. If 3a + 5b = 10 and 5a + 3b = 30, what is the average of a and b?   

3 5 10

5 3 30

a b

a b

8

5

2

8 8 40 a b

5 a b

5

2 2

a b

III. WORD PROBLEMS

READ, READ, READ, READ, and READ AGAIN!!!

A.) Strategies 

1.) Substitution i.e., “Plugging it in” 

Why???- Numbers make more sense than letters.- Choose numbers easy to work with, but not 0 and 1.- 2,3,5, etc. are good choices for algebra problems.- Multiples of 100 for percent problems.- Multiples of 60 for time problems.

When??? -You have NO idea how to do the problem-There is a variable in the question and the answers are all numbers-The problem is about “some number” and you have no clue as to what that number is.

B.) Examples Ex. The price of an item in a store is d dollars. If the sales tax on the item is s%, what is the total cost of x such items, including tax? 

a.)

b.)

c.)

  d.)

e.) 

1

100

xd s

100 ( )x d ds

( 100)

100

xd s

1xds

xds

Let’s choose some numbers for d, s, and x.   The total price for 1 item =

The total price for 10 items =

Which choice gives us $105.00? – Start with A

 Obviously, B.) is out

10

5%

10

d

s

x

10 10(.05) $10.50

10 $10.50 $105.00

xds 10 10 5 500 NO

C.)   D.)

E.)

 

100 ( )x d ds

10 10 5 16

100

1

100

xd s NO

100 10 10 10 5 25000 NO

( 100)

100

xd s 10 10 (5 100)105

100

YES!!!

Ex. Vehicle A travels at x miles per hour for x hours. Vehicle B travels a miles faster than Vehicle A, and travels b hours longer than Vehicle A. Vehicle B travels how much farther than Vehicle A, in miles? 

a.)

b.)

c.)

  d.)

e.) 

ax bx ab

2x abx ab

22x a b x ab

2x ab

2 2a b

Let’s choose some numbers for x, a, and b.   Vehicle A =

Vehicle B =

Vehicle B – Vehicle A =

 By substitution – A.)

20

10

5

x

a

b

20 20 400

220 10(5) 350 2x ab

30 25 750

750 400 350

The End!!!