1. 8.2 MULTIPLYING and FACTORING POLYNOMIALS: Distributive Property: for any real numbers a, b, and c : a(b+c) = (b+c)a = ab + ac. Greatest Common Factor (GCF): The number, variable and/or term that each monomial or polynomial have in common.

2. GOAL:

3. MULTIPLYING POLYNOMIALS: When multiplying polynomials we must keep in mind the laws of exponents from chapter 7 and the distributive property. –2x 2 (7x 3 +5x 2 – 12) Ex: What is the simplest for of:

4. MULTIPLYING POLYNOMIALS: Use the distributive property and exponent rules: –2x 2 (7x 3 +5x 2 – 12) (–2)(7)x 3 x 2 +(–2)(5)x 2 x 2 – (–2)(12) x 2 –14x 5 –10x 4 + 24x 2

5. REAL-WORLD (x): We multiply polynomials using distributive property and exponential rules: Ex:A rectangular wooden frame has side lengths of 5x and 7x+1. The rectangular opening for a picture has side lengths of 3x and 5x. What is the area of the wooden part of the frame?

6. SOLUTION: A rectangular wooden frame : 5x 7x+1 Area = b.h Area = (7x+1)(5x) Area of outside = 35x 2 + 5x

7. SOLUTION: The rectangular opening for a picture has side 3x 5x Area = b.h Area = (5x)(3x) Area = 15x 2

8. SOLUTION: What is the area of the wooden part of the frame? Area of inside : - 15x 2 Area of wood part:  20x 2 + 5x 5x 7x+1 3x 5x Area of outside: 35x 2 + 5x

9. FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: Ex: What is the GCF of : 1) 5X 3, 25x 2 and 45x? 2) 9x 6 +15x 4 +12x 2 3) p(p + 2) – 5(p + 2)

10. FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: 1) 5x 3, 25x 2 and 45x? 5x 3  5∙x∙x∙x 25x 2  5∙5∙x∙x 45x  3∙3∙5∙x In common: 5 x The GCF is 5x.

11. FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: 1) 9x 6 +15x 4 +12x 2 9x 6  3∙ 3∙x∙x∙x∙x∙x∙x 15x 4  3∙5∙x∙x∙x∙x 12x 2  3∙4∙x∙x In common: 3xx = 3x 2 The Factored form is: 3x 2 (3x 4 +5x 2 +4)

12. FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: 3) p(p + 2) – 5(p + 2) p(p + 2)  p∙(p + 2) – 5(p + 2)  – 5∙(p + 2) In common: (p + 2) The Factored form is: (p + 2)(p– 5)

13. VIDEOS: Polynomials Multiplying Multiplying: https://www.khanacademy.org/math/trigonometry/polyn omial_and_rational/polynomial_tutorial/v/multiplying- polynomials

14. VIDEOS: Greatest Common Factor https://www.khanacademy.org/math/algebra/polyno mials/multiplying_polynomials/v/monomial-greatest- common-factor https://www.khanacademy.org/math/algebra/polynomials/ multiplying_polynomials/v/factoring-and-the-distributive- property-2

15. CLASSWORK: Page 477-478: Problems: As many as needed to master the concept