1. Section 10.2 What we are Learning: To use the GCF and the distributive property to factor polynomials To use grouping techniques to factor polynomials with four or more terms

2. 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: x 2 + 75x Factored Form: x(x + 75) – Using distribution you will end up back at x 2 + 75

3. 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

4. Distribution Example: Factor 12mn 2 – 18m 2 n 2 12mn 2 122623mn 2 mnn 18m 2 n 2 182933m2n2m2n2 mmnn Our GCF is 6mn 2 We have 2 left over from 12mn 2 We have 3m left over from 18m 2 n 2 6mn 2 (2 – 3m)

5. 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

6. Example: Factor 20abc + 15a 2 c – 5ac (20abc + 15a 2 c – 5ac) What is common? 5, a, c 5ac(4b + 3a – 1) Do I have anything common still?

7. 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

8. 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

9. 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 common 3a(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)

10. 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)

11. 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)

12. Let’s Work These Together: Factor each polynomial: 9t 2 + 36t 15xy 3 + y 4

13. Let’s Work These Together: Factor each polynomial 2ax + 6xc + ba + 3bc 6a 2 – 6ab + 3bc – 3ca

14. Let’s Work These Together: 3m 2 – 5m 2 p + 3p 2 – 5p 3 5a 2 – 4ab +12b 3 – 15ab 2 Factor each polynomial

15. Homework: Page 570 – 33 to 49 odd