7.8 Summary of Factoring

CORD Math Mrs. Spitz Fall 2006

Objective: • Factor polynomials by applying various methods of factoring.

Assignment • Pg. 288 #5-50 all

Introduction • In this chapter, you have used various methods to factor different types of polynomials. The following chart on the next slide summarizes these methods and can help you decide when to use a specific method.

Check for: Greatest Common Factor Difference of squares Perfect Square trinomial Trinomial w/ two binomial factors Pairs of terms w/ common monomial factor Summary Chart Two X Number of Terms Three Four + X X X X X X X

Watch . . . • Whenever there is a GCF other than 1, always factor it out first. Then check the appropriate factoring methods in the order shown in the table. Use these methods to factor until all of the factors are prime.

Ex. 1: Factor 3x 2 - 27 • First check for a GCF. Then since there are two terms, check for the difference of squares.

3

x

2  27 3 is the GCF   3 (

x

2 3 (

x

 9 )  3 )(

x

 3 ) x 2 – 9 is the difference of squares since x  x = x 2 and 3  3 = 9 • Thus 3x 2 – 27 is completely factored as 3(x +3)(x – 3)

Ex. 2: Factor 9y 2 – 58y + 49 • The polynomial has three terms. So check for the following: 1. GCF – The GCF is 1 2. Perfect square trinomial – Although 9y 2 = (3y) 2 and 49 = (7) 2 , 58y ≠ 2(3y)(7).

3. Trinomial w/ two binomial factors: Are there two numbers whose product is 9  49 or 441 and whose sum is -58?

Ex. 2: Factor 9y2 – 58y + 49 • You must find two numbers whose product is 9 · 49 or 441 and whose sum is -58.

Factors of 441 -1, -441 -3, -147 -7, -63 -9, -49 -21, -21 Sum of factors -1 + -441=-442 -3 + -147=-150 -7 + -63 = -70 -9 + -49 = -58 -21 + -21 = -42

• • Ex. 2: Factor 9y 2 – 58y + 49 Trinomial w/ two binomial factors: Are there two numbers whose product is 9  49 or 441 and whose sum is -58? YES, the product of -9 and -49 is 441 and their sum is -58. -9 -58 -49  9   9 49 9  1  (

y

 1 ) 1  ( 9

y

 49 ) 441 (9y – 49)(y – 1) Check this by using FOIL.

Ex. 3 – Factor by grouping 4

m

4

n

 6

m

3

n

2  16

m

2

n

 24

mn

2  2

mn

( 2

m

3  3

m

2

n

 8

m

 12

n

) Pull GCF of 2mn  2

mn

[( 2

m

3  3

m

2

n

)  (  8

m

 12

n

)] Group two commons together  2

mn

[

m

2 ( 2

m

 3

n

)  4 ( 2

m

 3

n

)] Pull m 2 out of first group and -4 out of 2 nd group.

 2

mn

[(

m

2  4 )( 2

m

 3

n

)] Recognize Difference of squares  2

mn

[(

m

 2 )(

m

 2 )( 2

m

 3

n

)] Factor remaining and simplify.

You can check this by multiplying the factors.

Chalkboard examples #1 • Factor 5ax 2 – 45a 5

ax

2  45

a

5

a

(

x

2  9 ) 5

a

(

x

 3 )(

x

 3 )

Chalkboard examples #2 • Factor 32x 3 – 50x 32

x

3  50

x

2

x

(

x

2  25 ) 2

x

(

x

 5 )(

x

 5 )

Chalkboard examples #3 • Factor x 4 + x 3 – 12x 2

x

4 

x

3  12

x

2

x

2 (

x

2 

x

 12 )

x

2 (

x

 4 )(

x

 3 ) GCF of x 2 Factors of -12 that subtract to give you 1.

Chalkboard examples #4 • Factor 12a 3 – 16a 2 – 16a 12

a

3  16

a

2  16

a

4

a

( 3

a

2  4

a

 4 ) 4

a

( 3

a

 2 )(

a

 2 ) GCF of x 2 Factors of -12 that subtract to give you -4.

• • CB ex. 4: Factor 3a 2 – 4a – 4 Trinomial w/ two binomial factors: Are there two numbers whose product is 3  -4 or -12 and whose sum is -4? YES, the product of -6 and 2 is -12 and their sum is -4.  6 3   2 1  (

a

 2 ) -4 -6 +2 2 3  ( 3

a

 2 ) -12 (a - 2)(3a + 2) Check this by using FOIL.