9.2

1. 9.2 * & Factoring Polynomials

2. 9.2 - * & Factoring Polys. • Goals / “I can….” • Multiply a polynomial by a monomial • Factor a monomial from a polynomial

3. 9.2 - * & Factoring Polys. • To multiply a polynomial by a monomial, you can use two methods. 1. Box Method 2. Distribution.

4. 9.2 - * & Factoring Polys. • Distribution – Use the rainbows of death for each term 8m (m + 6)

5. 9.2 - * & Factoring Polys. • Simplify the product -p (p – 11)

6. 9.2 - * & Factoring Polys. • Simplify the product 5x (x - 2x + 5) 2

7. 9.2 - * & Factoring Polys. • Simplify the product 2g (4g + 3g + 2) = 3 2

8. 9.2 - * & Factoring Polys. • Write in standard form 3x ( 4x - 5x + 9) 2

9. 9.2 - * & Factoring Polys. • Finding the Greatest Common Factor (GCF) • The GCF is the term that can go into EVERY term in a polynomial.

10. 9.2 - * & Factoring Polys. Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. • Finding the GCF of a List of Integers or Terms • Prime factor the numbers. • Identify common prime factors. • Take the product of all common prime factors. • If there are no common prime factors, GCF is 1.

11. 9.2 - * & Factoring Polys. Example Find the GCF of each list of numbers. • 12 and 8 12 = 2· 2· 3 8 = 2·2· 2 So the GCF is 2·2 = 4. • 7 and 20 7 = 1 · 7 20 = 2 · 2 · 5 There are no common prime factors so the GCF is 1.

12. 9.2 - * & Factoring Polys. Example Find the GCF of each list of numbers. • 6, 8 and 46 6 = 2 · 3 8 = 2· 2 · 2 46 = 2· 23 So the GCF is 2. • 144, 256 and 300 144 = 2 ·2 ·2 · 3 · 3 256 = 2·2 ·2 · 2 · 2 · 2 · 2 · 2 300 = 2·2 · 3 · 5 · 5 So the GCF is 2·2 = 4.

13. 9.2 - * & Factoring Polys. • x3 and x7 x3 = x ·x·x x7 = x ·x·x·x ·x·x·x So the GCF is x · x· x = x3 • 6x5 and 4x3 6x5 = 2 · 3 · x · x· x 4x3 = 2 · 2 ·x ·x·x So the GCF is 2·x ·x·x = 2x3 Example Find the GCF of each list of terms.

14. 9.2 - * & Factoring Polys. Example Find the GCF of the following list of terms. • a3b2, a2b5 and a4b7 a3b2 = a ·a·a· b· b a2b5 = a · a· b· b · b· b· b a4b7 = a · a· a· a· b· b · b· b· b· b· b • So the GCF is a · a· b· b = a2b2 Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable.

15. 9.2 - * & Factoring Polys. The first step in factoring a polynomial is to find the GCF of all its terms. Then we write the polynomial as a product by factoring out the GCF from all the terms. The remaining factors in each term will form a polynomial.

16. 9.2 - * & Factoring Polys. Example Factor out the GCF in each of the following polynomials. 1)6x3 – 9x2 + 12x = 3· x· 2 ·x2 – 3·x· 3 ·x + 3·x· 4 = 3x(2x2 – 3x + 4) 2)14x3y + 7x2y – 7xy = 7 ·x·y· 2 ·x2 + 7·x·y· x – 7·x·y· 1 = 7xy(2x2 + x – 1)

17. 9.2 - * & Factoring Polys. Example Factor out the GCF in each of the following polynomials. • 1)6(x + 2) – y(x + 2) = • 6 ·(x + 2) – y·(x + 2) = (x + 2)(6 – y) • 2)xy(y + 1) – (y + 1) = xy·(y + 1) – 1 ·(y + 1) = (y + 1)(xy – 1)

18. 9.2 - * & Factoring Polys. • Example: 2x + 10x + 6x • What are the common factors that go into 2, 10 and 6? • What are the common factors that go into x , x , and x ? • So the GCF is 2 4 3 3 4 2

19. 9.2 - * & Factoring Polys. • Factor a monomial • Find the GCF, and un-distribute it form the polynomial. • Example: 4x + 8x + 12x The GCF is 3 2

20. 9.2 - * & Factoring Polys. • Find the GCF 3x - 12x + 15x 3 2

21. 9.2 - * & Factoring Polys. • Find the GCF 8x - 12x 2

22. 9.2 - * & Factoring Polys. • Find the GCF 36x + 24