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Chapter 6 Section 2
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factoring Trinomials
Factor trinomials with a coefficient of 1 for the second-degree term.
Factor such trinomials after factoring out the greatest common factor.
6.2
2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factoring Trinomials
Using the FOIL method, we see that the product of the binomial k − 3 and k +1 is
(k − 3)(k + 1) = k2 − 2k − 3. Multiplying
Suppose instead that we are given the polynomial k2 − 2k − 3 and want to rewrite it as the product (k − 3)(k + 1). That is,
k2 − 2k − 3 = (k − 3)(k + 1). Factoring
Recall from Section 6.1 that this process is called factoring the polynomial. Factoring reverses or “undoes” multiplying.
Slide 6.2-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Factor trinomials with a coefficient of 1 for the second-degree term.
Slide 6.2-4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor trinomials with a coefficient of 1 for the second-degree term.
When factoring polynomials with integer coefficients, we use only integers in the factors. For example, we can factor x2 + 5x + 6 by finding integers m and n such that
x2 + 5x + 6 = (x + m)(x + n).
Comparing this result with x2 + 5x + 6 shows that we must find integers m and n having a sum of 5 and a product of 6.
2 25 6 .x nx xm mnx
Sum of m and n is 5.
Product of m and n is 6.
2 .m n n m mx x x x nx 2 .n m mnx x
To find these integers m and n, we first use FOIL to multiply the two binomials on the right side of the equation:
Slide 6.2-5
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Since many pairs of integers have a sum of 5, it is best to begin by listing those pairs of integers whose product is 6. Both 5 and 6 are positive, so consider only pairs in which both integers are positive.
Both pairs have a product of 6, but only the pair 2 and 3 has a sum of 5. So 2 and 3 are the required integers, and
x2 + 5x + 6 = (x + 2)(x + 3).
Check by using the FOIL method to multiply the binomials. Make sure that the sum of the outer and inner products produces the correct middle term.
Slide 6.2-6
Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor y2+ 12y + 20.
Solution:
10 2y y
Factors of 20 Sums of Factors
1, 20 1 + 20 = 21
2, 10 2 + 10 = 12
4, 5 4 + 5 = 9
You can check your factoring by graphing both the unfactored and factored forms of polynomials on your graphing calculators.
Slide 6.2-7
EXAMPLE 1 Factoring a Trinomial with All Positive Terms
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Factor y2 − 10y + 24.
Solution:
6 4y y
Factors of 24 Sums of Factors
− 1 , −24 −1 + (−24) = −25
−2 , −12 −2 + (−12) = −14
−3 , −8 −3 + (−8) = −11
−4 , −6 −4 + (−6) = −10
Slide 6.2-8
EXAMPLE 2 Factoring a Trinomial with a Negative Middle Term
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor z2 + z − 30.
Solution:
6 5z z
Factors of − 30 Sums of Factors
− 1 , 30 −1 + (30) = 29
1 , − 30 1 + (−30) = −29
5 , − 6 5 + (− 6) = −1
−5 , 6 −5 + (6) = 1
Slide 6.2-9
EXAMPLE 3 Factoring a Trinomial with a Negative Last (Constant) Term
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor a2 − 9a − 22.
Solution:
11 2a a
Factors of −22 Sums of Factors
−1 , 22 −1 + 22 = 21
1, −22 1 + (−22) = −21
−2 , 11 −2 + 11 = 9
2 , −11 2 + (−11) = −9
Slide 6.2-10
EXAMPLE 4 Factoring a Trinomial with Two Negative Terms
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Some trinomials cannot be factored by using only integers. We call such trinomials prime polynomials.
Slide 6.2-11
Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d)
Summarize the signs of the binomials when factoring a trinomial whose leading coefficient is positive.
1. If the last term of the trinomial is positive, both binomials will have the same “middle” sign as the second term.
2. If the last term of the trinomial is negative, the binomials will have one plus and one minus “middle” sign.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor if possible.
Factors of 14 Sums of Factors
−1 , −14 −1 + (−14) =
−15
−2 , −7 −2 + (−7) = −9
Solution:
2 8 14m m
2 2y y
Prime
Prime
Factors of 2 Sums of Factors
1, 2 1 + 2 = 3
Slide 6.2-12
EXAMPLE 5 Deciding Whether Polynomials Are Prime
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Guidelines for Factoring x2 + bx + c
Find two integers whose product is c and whose sum is b.
1. Both integers must be positive if b and c are positive.
2. Both integers must be negative if c is positive and b is negative.
3. One integer must be positive and one must be negative if c is negative.
Slide 6.2-13
Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor r2 − 6rs + 8s2.
Solution:
4 2r s r s
Factors of 8s2 Sums of Factors
−1s , −8s − 1s + (−8s) = −9s
−2s , −4s −2s + (−4s) = −6s
Slide 6.2-14
EXAMPLE 6 Factoring a Trinomial with Two Variables
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Factor such trinomials after factoring out the greatest common factor.
Slide 6.2-15
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
If a trinomial has a common factor, first factor it out.
Slide 6.2-16
Factor such trinomials after factoring out the greatest common factor.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor 3x4 − 15x3 + 18x2.
2 23 5 6x x x
23 3 2x x x
Solution:
When factoring, always look for a common factor first. Remember to include the common factor as part of the answer. As a check, multiplying out the factored form should always give the original polynomial.
Slide 6.2-14
EXAMPLE 7 Factoring a Trinomial with a Common Factor
