6 – 4: 6 – 4: Factoring and SolvingFactoring and SolvingPolynomial EquationsPolynomial Equations

(Day 1)(Day 1)

Objective: Factor polynomial expressions.Use factoring to solve polynomial expressions.

Example 1: Factor the Trinomial

A ) 3x3 – 3x2 – 18xGCF = 3x3x (x2 – x – 6)3x (x-3)(x+2)

B ) 80x4y + 44x3y2 - 16x2y3

GCF = 4x2y4x2y(20x2 + 11xy – 4y2)

4x2y(5x + 4y)(4x - y)

Special Factoring Patterns:1. Sum of two cubes

3 3 2 2

3 28 2 2 4

a b a b a ab b

x x x x

2. Difference of two cubes

3 3 2 2

3 28 1 2 1 4 2 1

a b a b a ab b

x x x x

Example 2:Factor each polynomial

A. x3 + 27(x + 3) (x2 – 3x + 9)

B. 16u5 – 250u2

GCF: 2u2

2u2(8u3 – 125)2u2(2u-5)(2u2 + 10 u – 25

For some polynomials you can factor by grouping pairs of terms that have a

common monomial factor. The pattern for this is as follows.

ra rb sa sb r a b s a b

r s a b

Example 4: Factor the polynomialA. 7x3 – 7x2 + x - 1

(7x3 – 7x2) + (x – 1)7x2(x – 1) + 1(x – 1)

(7x2 + 1)(x – 1)

B. 3x6 – 3x4 + 2x3 – 2x(3x6 – 3x4) + (2x3 – 2x)

3x4(x2 - 1) + 2x(x2 – 1)(3x4 + 2x)(x2 – 1)

x(3x3 + 2)(x2 – 1)

HOMEWORK

AM Objectives:• #3 Factor Trinomials

• #4: Factor Sum or Difference of 2 Cubes

Tomorrow’s WorkAM Objectives:

#83 Factor Polynomials into Binomials & Trinomials

#84 Factor by Grouping

Example 3: Factor the polynomial 3 22 9 18x x x

3 2 2

2

2 9 18 2 9 2

9 2

3 3 2

x x x x x x

x x

x x x

An expression in the form 2au bu c

where u is any expression in x is said to be in quadratic form. The factoring techniques for quadratics (factoring, completing the square, and the quadratic formula) can be used to factor such expressions.

Example 3Factor each polynomial

4 6 4 2a. 81 16 b. 4 20 24x x x x

24 2 2

2 2

2

a. 81 16 9 4

9 4 9 4

3 2 3 2 9 4

x x

x x

x x x

6 4 2 2 4 2

2 2 2

b. 4 20 24 4 5 6

4 3 2

x x x x x x

x x x