CHAPTER

5Polynomials: Factoring

Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

5.1 Introduction to Factoring

5.2 Factoring Trinomials of the Type x2 + bx + c

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

5.4 Factoring ax2 + bx + c, a ≠ 1: The ac-Method

5.5 Factoring Trinomial Squares and Differences of Squares

5.6 Factoring: A General Strategy

5.7 Solving Quadratic Equations by Factoring

5.8 Applications of Quadratic Equations

OBJECTIVES

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

The FOIL Method

(continued)

Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

The FOIL Method

Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution 1. First, check for a common factor. There is none other than 1 or 1.2. Find the First terms whose product is 3x2. The only possibilities are 3x and x:

(3x + )(x + )

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

A Factor: 3x2 14x 5

(continued)

Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution 3. Find the Last terms whose product is 5. Possibilities are (5)(1), (5)(1)

Important!: Since the First terms are not identical, we must also consider the above factors in reverse order: (1)(5), and (1)(5).

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

A Factor: 3x2 14x 5

(continued)

Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

4. Knowing that the First and Last products will check, inspect the Outer and Inner products resulting from steps (2) and (3) Look for the combination in which the sum of the products is the middle term

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

A Factor: 3x2 14x 5

(continued)

Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE(3x 5)(x + 1) = 3x2 + 3x 5x 5

= 3x2 2x 5 (3x 1)(x + 5) = 3x2 + 15x x 5

= 3x2 + 14x 5(3x + 5)(x 1) = 3x2 3x + 5x 5

= 3x2 + 2x 5(3x + 1)(x 5) = 3x2 15x + x 5

= 3x2 14x 5

Wrong middle term

Wrong middle term

Wrong middle term

Correct middle term!

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

A Factor: 3x2 14x 5

Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution 1. First, factor out the largest common factor, 3x:

3x(6x2 47x 8)2. Factor 6x2 47x 8. Since 6x2 can be factored as 3x 2x or 6x x, we have two possibilities

(3x + )(2x + ) or (6x + )(x + )

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

B Factor: 18x3 141x2 24x

(continued)

Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution 3. There are several pairs of factors of 8. List each way:

8, 1 1, 8 2, 4 4, 2 8, 1 1, 8 2, 4 4, 2

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

B Factor: 18x3 141x2 24x

(continued)

Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLETrial Product(6x + 4)(x 2) 6x2 12x + 4x 8

= 6x2 8x 8(6x 4)(x + 2) 6x2 + 12x 4x 8

= 6x2 + 8x + 8(6x + 1)(x 8) 6x2 48x + x 8

= 6x2 47x 8We do not need to consider (3x + )(2x + ). The complete factorization is 3x(6x + 1)(x 8).

Wrong middle term

Wrong middle term

Correct middle term

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

B Factor: 18x3 141x2 24x

Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

STUDY TIPTips for Factoring ax2 + bx + c

Always factor out the largest common factor, if one exists.

Once the largest common factor has been factored out of the original trinomial, no binomial factor can contain a common factor (other than 1 or –1).

If c is positive, then the signs in both binomial factors must match the sign of b.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

STUDY TIP

Reversing the signs in the binomials reverses the sign of the middle term of their product.

Organize your work so that you can keep track of which possibilities you have checked.

Always check by multiplying.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution: An important problem-solving strategy is to find a way to make problems look like problems we already know how to solve. Rewrite the equation in descending order.

14x + 5 3x2 = 3x2 + 14x + 5Factor out the 1:

3x2 + 14x + 5 = 1(3x2 14x 5) = 1(3x + 1)(x 5)

The factorization of 14x + 5 3x2 is 1(3x + 1)(x 5).

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

C Factor: 14x + 5 3x2

Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution: No common factors exist, we examine the first term, 6x2. There are two possibilities:

(2x + )(3x + ) or (6x + )(x + ).The last term 12y2, has the following pairs of factors:

12y, y 6y, 2y 4y, 3yand 12y, y 6y, 2y 4y, 3yas well as each of the pairings reversed.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

D Factor: 6x2 xy 12y2

(continued)

Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE

Some trials like (2x 6y)(3x + 2y) and (6x + 4y)(x 3y), cannot be correct because (2x 6y) and (6x + 4y) contain a common factor, 2.

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

D Factor: 6x2 xy 12y2

(continued)

Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLETrial Product(2x + 3y)(3x 4y) 6x2 8xy + 9xy 12y2

= 6x2 + xy 12y2

Our trial is incorrect, but only because of the sign of the middle term. To correctly factor, simply change the signs in the binomials.(2x 3y)(3x + 4y) 6x2 + 8xy 9xy 12y2

= 6x2 xy 12y2

The correct factorization is (2x 3y)(3x + 4y).

5.3 Factoring ax2 + bx + c, a ≠ 1: The FOIL Method

a Factoring ax2 + bx + c, a ≠ 1, using the FOIL method.

D Factor: 6x2 xy 12y2

Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.