Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 1 Chapter 6 Polynomial...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 1

Chapter 6Polynomial Functions


6.4 Factoring Trinomials of the Form x2 + bx + c; Factoring Out the GCF


Comparing Multiplying with Factoring

Multiplying and factoring are reverse processes. For example,


Factoring a Trinomial of the Form x2 + bx + c

To see how to factor x2 + 5x + 6, let’s take another look at how we find the product (x + 2)(x + 3):


Example: Factoring a Trinomial of the Form x2 + bx + c

Factor x2 + 11x + 24.


Solution

We need two integers whose product is 24 and whose sum is 11.

Since 3(8) = 24 and 3 + 8 = 11, we conclude that the last terms of the factors are 3 and 8.


Solution

x2 + 11x + 24 = (x + 3)(x + 8)

Check by finding the product of the result:

(x + 3)(x + 8) = x2 + 8x + 3x + 24 = x2 + 11x + 24

By the commutative law, (x + 3)(x + 8) = (x + 8)(x + 3),

so we can write the factors in either order.


Factoring x2 + bx + c

To factor x2 + bx + c, look for two integers p and q whose product is c and whose sum is b. That is pq = c and p + q = b. If such integers exist, the factored polynomial is

(x + p)(x + q)


Factoring x2 + bx + c with c Positive

To factor a trinomial of the form x2 + bx + c with a positive constant term c,

• If b is positive, look for two positive integers whose product is c and whose sum is b. For example,


Factoring x2 + bx + c with c Positive

• If b is negative, look for two negative integers whose product is c and whose sum is b. For example,


Example: Factring a Trinomial of the Form x2 + bx + c

Factor w2 – 3w – 18.


Solution

We need two integers whose product is –18 and whose sum is –3. Since the product is negative, the two integers must have difference signs. Here are the possibilities:


Solution

Since 3(–6) = –18 and 3 + (–6) = –3, we conclude that the last terms of the factors are 3 and –6.

(w + 3)(w – 6)

Check by finding the product:

(w + 3)(w – 6) = w2 – 6w + 3w – 18 = w2 – 3w – 18


Factoring x2 + bx + c with c Negative

To factor a trinomial of the form x2 + bx + c with a negative constant term c, look for two integers with different signs whose product is c and whose sum is b. For example,


Example: Factoring a Trinomial with Two Variables

Factor a2 + 6ab + 8b2.


Solution

Write the trinomial in the form a2 + (6b)a + 8b2. We need two monomials whose product is 8b2 and whose sum is 6b. So, the last two terms are 2b and 4b.

a2 + 6ab + 8b2 = (a + 2b)(a + 4b)

Check by finding the product.

(a + 2b)(a + 4b) = a2 + 6ab + 8b2


Prime Polynomials

Just as a prime number has no positive factors other than itself and 1, a polynomial that cannot be factored is called prime.


Example: Identifying a Prime Polynomial

Factor –14 + 6x + x2.


Solution

Write the polynomial in descending order:

x2 + 6x – 14


SolutionWe need two integers whose product is –14 and whose sum is 6. Since the product is negative, the integers must have different signs. Here are the possibilities:

Because none of the sums equal 6, we conclude that the trinomial is prime.


Factoring Out the GCF

Definition

The greatest common factor (GCF) of two or more terms is the monomial with the largest coefficient and the highest degree that is a factor of all the terms.


Example: Factoring Out the GCF

Factor 18x4 – 30x2.


Solution

Begin by factoring 18x4 and 30x2:

18x4 = 2 ∙ 3 ∙ 3 ∙ x ∙ x ∙ x ∙ x30x2 = 2 ∙ 3 ∙ 5 ∙ x ∙ x

There are four common factors, shown in blue. So, the GCF is 6x2:

2 224 218 30 6 63 5xx x xx 2 236 5x x


Solution

Use a graphing calculator table to verify our work.


Example: Completely Factoring a Polynomial

Factor 3x3 + 21x2 + 36x.


Solution

The GCF is 3x:

3x3 + 21x2 + 36x = 3x(x2 + 7x + 12)


SolutionTemporarily put aside the GCF, 3x.

To factor x2 + 7x + 12, we need two integers whose product is 12 and whose sum is 7:

Because 3(4) = 12 and 3 + 4 = 7, we conclude that the last terms of the factors are 3 and 4.


Solution

2 7 12 ( 3 4)( )x x x x

So,

3 2 23 21 36 3 7 12 3 ( 3)( 4)x x x x x x x x x


Factor Out the GCF First

In general, when the leading coefficient of a polynomial is positive and the GCF is not 1, first factor out the GCF.

But, don’t forget to include the GCF as part of your answer!


Factoring Completely

Warning

When factoring a polynomial, always completely factor it.


Factoring when the Leading Coefficient is Negative

When the leading coefficient of a polynomial is negative, first factor out the opposite of the GCF.


Example: Factoring Out the Opposite of the GCF

Factor –2r4 + 18r3 – 40r2.


Solution

For this polynomial, the GCF is 2r2. The leading coefficient of –2r4 + 18r3 – 40r2 is –2, which is negative. So, first factor out the opposite of the GCF:

4 3 2 2 22 18 40 2 9 20r r r r r r 22 ( 5)( 4)r r r


Solution

Use a graphing calculator table to verify our work.


Summary

1. If the leading coefficient of a polynomial is positive and the GCF is not 1, first factor out the GCF. If the leading coefficient is negative, first factor out the opposite of the GCF.

2. Always completely factor a polynomial.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.4, Slide 1 Chapter 6 Polynomial...

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