Lesson 8-2Multiplying and

Factoring Polynomials

Multiplying a binomial by a monomial uses the Distribute property

Multiplying Polynomials

5(𝑥+5)

5(𝑥+5) Distribute the 5

(5 ∙𝑥 )+¿

5 𝑥+25(5 ∙5)

Multiplying Polynomials

−𝑥3 (9 𝑥4−2𝑥3+7 )=¿ ) (7)

What is the simpler form of

A

B

C

D

Solution:

Multiplying two binomial uses the FOIL

Multiplying Polynomials

(3 𝑥−6)(𝑥+5)

¿ (3 𝑥 ∙ 𝑥 )+ (3𝑥 ∙5 )

¿3 𝑥2−6 𝑥+15 𝑥−30

(3 𝑥−6)(𝑥+5)

− (6 ∙𝑥 )−(6 ∙5)

¿3 𝑥2+9 𝑥−30

First Outer Inner Last

Multiplying Polynomials

(𝑥+3)(𝑥+2)

¿ (𝑥 ∙ 𝑥 )+ (𝑥 ∙2 )

¿ 𝑥2+2𝑥+3 𝑥2+6 𝑥

(𝑥+3 𝑥)(𝑥+2)

+(3 𝑥 ∙ 𝑥 )+(3 𝑥 ∙2)

¿ 4 𝑥2+8 𝑥

What is the simpler form of

Multiplying Polynomials

(𝑎+𝑏 )2=(𝑎+𝑏)(𝑎+𝑏)

¿ (𝑎 ∙𝑎)+(𝑎 ∙𝑏)

¿𝑎2+𝑎𝑏+𝑏𝑎+𝑏2

(𝑎+𝑏)(𝑎+𝑏)

+(𝑏 ∙𝑎 )+(𝑏∙𝑏)

¿𝑎2+2𝑎𝑏+𝑏2

What is the simpler form of Special case – Square of a binomial

Factoring Polynomials

Factors: When an integer is written as a product of integers, each of the integers in the product is a factor of the original number.

25 𝑥2=5 ∙5 ∙𝑥 ∙ 𝑥

12=2∙2 ∙3

Factoring PolynomialsGreatest common factor – largest quantity that is a factor of all the integers or polynomials involved.

Find the GCF of each list of numbers.

1) 6, 8 and 46 6 = 2 · 3 8 = 2 · 2 · 246 = 2 · 23 So the GCF is 2

2) 144, 256 and 300144 = 2 · 2 · 2 · 23 · 3256 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2300 = 2 · 2 · 3 · 5 · 5 So the GCF is 2 · 2 = 4.

EXAMPLE

Factoring Polynomials

1) x3 and x7

x3 = x · x · xx7 = x · x · x · x · x · x · xSo the GCF is x · x · x = x3

2) 6x5 and 4x3

6x5 = 2 · 3 · x · x · x x · x4x3 = 2 · 2 · x · x · x So the GCF is 2 · x · x · x = 2x3

Find the GCF of each list of terms.

Factoring Polynomials

So the GCF is 5 · x or 5x

What is the GCF terms of

Factoring Polynomials

To factor a polynomial, find the greatest common factor (GCF) of the coefficients and constants and also the GCF of the variables.

Factoring a polynomial reverses the multiplication process. It is writing a polynomial as a product of polynomials.

Then write the polynomial as a product by factoring out the GCF from all the terms.

Factoring PolynomialsWhat is the factored form of

Step 1 – Factor each term

Step 3 - Factoring out of the polynomial

The GCF is Step 2 – Find the GCF

¿𝟒 𝒙 (𝒙𝟒−𝟔 𝒙𝟐+𝟐)

Factoring PolynomialsWhat is the factored form of

Step 1 – Factor each term

Step 3 - Factoring out of the polynomial

The GCF is Step 2 – Find the GCF

¿𝟑 𝒙 (𝟐 𝒙𝟐−𝟑 𝒙+𝟒)

Factoring Polynomials

Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial.

This will usually be followed by additional steps in the process.