Chapter 5

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-1

Factoring

2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-2

5.1 – Factoring a Monomial from a Polynomial

5.2 – Factoring by Grouping5.3 – Factoring Trinomials of the Form

ax2 + bx + c, a = 15.4 – Factoring Trinomials of the Form

ax2 + bx + c, a ≠ 15.5 – Special Factoring Formulas and a

General Review of Factoring5.6 – Solving Quadratic Equations Using

Factoring5.7 – Applications of Quadratic Equations

Chapter Sections

3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-3

Factoring Trinomials of

the Formax2 + bx + c, a =

1

4 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-4

Factoring TrinomialsRecall that factoring is the reverse process of multiplication. Using the FOIL method, we can show that

((xx + 3)( + 3)(xx + 4) = + 4) = xx22 + 7 + 7xx + 12. + 12.

Therefore xx22 + 7 + 7xx + 12 + 12 = = ((xx + 3)( + 3)(xx + 4) + 4) Note that this trinomial results in the product of two binomials whose first term is x and second term is a number (including its sign).

5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-5

Factoring TrinomialsFactoring any polynomial of the form Factoring any polynomial of the form xx22 + b + bxx + c + cwill result in a pair of binomials:will result in a pair of binomials:

Numbers go here.Numbers go here.

x2 + bx + c = (x +?)(x +?)

O

( x + 3 )( x + 4 )

F

I

L

= x2 + 4x + 3x + 12

= x2 + 7x + 12

F O I L

6 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-6

Factoring Trinomials1. Find two numbers whose product

equals the constant, c, and whose sum equals the coefficient of the x-term, b.

2. Use the two numbers found in step 1, including their signs, to write the trinomial in factored form. The trinomial in factored form will be

(x + first number) (x + second number)

7 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-7

Examplesa.) Factor x2 - 11x - 60.

x2 - 11x - 60 = (x + ?) (x + ?)Replace the ?s with two numbers that are the product of -60 and the sum of -11.x2 + 8x + 15 = (x -15) (x + 4)

b.) Factor x2 + 5x + 12.This is a prime polynomial

because it cannot be factored.

8 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 5-8

Examples Continuedc.) Factor x2 + 3xy + 2y2.

We must find two numbers whose product is 2 (from 2y2) and whose sum is 3 (from 3xy). The two numbers and 1 and 2. Thus,x2 + 3xy + 2y2 = (x + 1y)(x + 2y) =

(x + y)(x + 2y)