Intermediate Algebra (11th edition)

326 CHAPTER 6 Factoring

OBJECTIVES OBJECTIVE 1 Factor trinomials when the coefficient of the quadratic term

is 1.We begin by finding the product of and

FOIL

Combine like terms.

By this result, the factored form of is

Multiplication

Factored form Product

Factoring

Since multiplying and factoring are operations that “undo” each other, factoring

trinomials involves using FOIL in reverse. As shown here, the -term comes from

multiplying xand x, and comes from multiplying 3 and

Product of x and x is

Product of 3 and is.

We find the in by multiplying the outer terms, multiplying the

inner terms, and adding.

Outer terms:

Add to get

Inner terms:

Based on this example, use the following steps to factor a trinomial

where 1 is the coefficient of the quadratic term.

x^2 + bx+c,

3 #x= 3 x

1 x+ 321 x- 52 - 2 x.

x 1 - 52 =- 5 x

- 2 x x^2 - 2 x - 15

• 5 - 15

1 x+ 321 x- 52 =x^2 - 2 x- 15

x^2.

- 15 - 5.

x^2

1 x+ 321 x- 52 =x^2 - 2 x- 15

x^2 - 2 x - 15 1 x+ 321 x- 52.

= x^2 - 2 x- 15

= x^2 - 5 x+ 3 x- 15

1 x+ 321 x- 52

x+ 3 x-5.

Factoring Trinomials

6.2

1 Factor trinomials

when the

coefficient of the

quadratic term is 1.

2 Factor trinomials

when the coefficient

of the quadratic

term is not 1.

3 Use an alternative

method for

factoring trinomials.

4 Factor by

substitution.

Factoring

Step 1 Find pairs whose product is c.Find all pairs of integers whose product

is c,the third term of the trinomial.

Step 2 Find the pair whose sum is b.Choose the pair whose sum is b,the

coefficient of the middle term.

If there are no such integers, the trinomial cannot be factored.

x^2 bxc

A polynomial that cannot be factored with integer coefficients is a prime

polynomial.

x^2 + x+2, x^2 - x- 1, 2 x^2 + x+ 7 Examples of prime polynomials