Section 10.2

What we are Learning:To use the GCF and the distributive property to factor

polynomialsTo use grouping techniques to factor polynomials with

four or more terms

Factored Form of a Polynomial:

• A polynomial expressed as the product of monomials and polynomials– A completely factored form, – All GCF are accounted for

• Polynomial: x2 + 75x• Factored Form: x(x + 75)– Using distribution you will end up back at x2 + 75

Using Distribution to Factor Polynomials:

• First, find the GCF of each part of the polynomial

• Second, use the distributive property to express the polynomial as the product of the GCF and the remaining factor of each term

Distribution Example:• Factor 12mn2 – 18m2n2

12mn2

12

2 6

2 3

mn2

m n n

18m2n2

18

2 9

3 3

m2n2

m m n n

Our GCF is 6mn2

We have 2 left over from 12mn2

We have 3m left over from 18m2n2

6mn2(2 – 3m)

Another Way to Factor Polynomials:

• Draw parentheses around the polynomial• Ask yourself, “What is common in each term?”• Move the common parts outside of the

parentheses• Check to make sure you do not have any

common terms left inside the parentheses• This is your factored form

Example:

• Factor 20abc + 15a2c – 5ac(20abc + 15a2c – 5ac) What is common?

5, a, c5ac(4b + 3a – 1) Do I have anything common still?

Factoring by Grouping:

• Sometimes we have terms in a polynomial that have common factors in them

• We can use the associative property to group terms that have common factors

• Grouping the terms with common factors makes factoring simpler

We Can Factor by Grouping if…

• There are four or more terms• Terms with common factors can be grouped

together• The two factors are identical or differ by a

factor of -1

Grouping Example:• Factor 12ac + 21ad + 8bc + 14bd

• Notice that 12ac and 21ad have an “a” in common– Group these two together

• Notice that 8bc and 14bd have a “b” in common– Group these two together

(12ac + 21ad) + (8bc + 14bd) What is common3a(4c + 7d) + 2b(4c + 7d)

• Notice (4c + 7d) is a common factor• Group the terms on the outside of the parentheses

together

• (3a + 2b)(4c + 7d)

Recognizing Additive Inverses:

• Remember: the sum of additive inverses = 0• This means that (a – 3) is equivalent to (–a + 3)

• There will be times that we use additive inverses to factor polynomials

• In order to change one factor to the equivalent of the other multiply by (-1)

Additive Inverse Example:

• Factor 15x – 3xy + 4y – 20 Group these first

(15x – 3xy) + (4y – 20) 3x(5 – y) + 4( y – 5) Notice (5–y) and (y-5) are additive inverses

3x(-1)(5 – y) + 4(y – 5) Multiply one side by -1

-3x(y – 5) + 4(y – 5) Group the terms on the outside of parentheses

(-3x + 4)(y – 5)

Let’s Work These Together:

Factor each polynomial:• 9t2 + 36t • 15xy3 + y4

Let’s Work These Together:

Factor each polynomial• 2ax + 6xc + ba + 3bc • 6a2 – 6ab + 3bc – 3ca

Let’s Work These Together:

3m2 – 5m2p + 3p2 – 5p3 5a2 – 4ab +12b3 – 15ab2

Factor each polynomial

Homework:

• Page 570– 33 to 49 odd