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.

–2x2 (7x3+5x2 – 12)

Ex: What is the simplest for of:

MULTIPLYING POLYNOMIALS:Use the distributive property and exponent rules:

–2x2 (7x3+5x2 – 12)

(–2)(7)x3x2+(–2)(5)x2x2 – (–2)(12) x2

–14x5 –10x4 + 24x2

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

7x+1

Area = b.h

Area = (7x+1)(5x)

Area of outside = 35x2 + 5x

SOLUTION:The rectangular opening for a picture has side

3x 5x

Area = b.h

Area = (5x)(3x)

Area = 15x2

SOLUTION:What is the area of the wooden part

of the frame?

Area of inside : - 15x2 Area of wood part: 20x2 + 5x

5x

7x+1

3x 5x

Area of outside: 35x2 + 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? 2) 9x6+15x4+12x2

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:

5 x

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∙ ∙ ∙ ∙ ∙ ∙ ∙

15x4 3 5 x x x x ∙ ∙ ∙ ∙ ∙12x2 3 4 x x ∙ ∙ ∙

In common:

3xx = 3x2

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:PolynomialsMultiplying

Multiplying:

https://www.khanacademy.org/math/trigonometry/polynomial_and_rational/polynomial_tutorial/v/multiplying-polynomials

VIDEOS:Greatest Common Factor

https://www.khanacademy.org/math/algebra/polynomials/multiplying_polynomials/v/monomial-greatest-common-factor

https://www.khanacademy.org/math/algebra/polynomials/multiplying_polynomials/v/factoring-and-the-distributive-property-2

CLASSWORK:

Page 477-478:

Problems: As many as needed to master the concept