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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. GOAL: MULTIPLYING POLYNOMIALS: When multiplying polynomials we must keep in mind the laws of exponents from chapter 7 and the distributive property. Ex: What is the simplest for of: 2 –2x 3 2 (7x +5x – 12) MULTIPLYING POLYNOMIALS: Use the distributive property and exponent rules: –2x2 (7x3+5x2 – 12) (–2)(7)x3x2+(–2)(5)x2x2 – (–2)(12) x2 5 –14x 4 –10x + 2 24x 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? SOLUTION: A rectangular wooden frame : 5x Area = b.h 7x+1 Area = (7x+1)(5x) Area of outside = 2 35x + 5x SOLUTION: The rectangular opening for a picture has side Area = b.h 3x Area = (5x)(3x) 5x 2 Area = 15x SOLUTION: What is the area of the wooden part of the frame? 5x 3x 5x 7x+1 Area of outside: 35x2 + 5x 2 Area of inside : - 15x Area of wood part: 20x2 + 5x FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: Ex: What is the GCF of : 1) 5X3, 25x2 and 45x? 6 4 2 2) 9x +15x +12x 3) p(p + 2) – 5(p + 2) FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: 1) 5x3, 25x2 and 45x? 5x3 5∙x∙x∙x 25x2 5∙5∙x∙x 45x 3∙3∙5∙x In common: 5x The GCF is 5x. FINDING THE GCF: Remember that the Greatest Common Factor (GCF) is what they have in common: 1) 9x6+15x4+12x2 9x6 3∙ 3∙x∙x∙x∙x∙x∙x In common: 15x4 3∙5∙x∙x∙x∙x 3xx = 3x2 2 12x 3∙4∙x∙x The Factored form is: 3x2(3x4+5x2+4) 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) VIDEOS: Polynomials Multiplying Multiplying: https://www.khanacademy.org/math/trigonometry/polyn omial_and_rational/polynomial_tutorial/v/multiplyingpolynomials VIDEOS: Greatest Common Factor https://www.khanacademy.org/math/algebra/polyno mials/multiplying_polynomials/v/monomial-greatestcommon-factor https://www.khanacademy.org/math/algebra/polynomials/ multiplying_polynomials/v/factoring-and-the-distributiveproperty-2 CLASSWORK: Page 477-478: Problems: As many as needed to master the concept