Factoring Trinomials
1.Factor trinomials with a coefficient of 1 for the squared term.
2.Factor such trinomials after factoring out the greatest common factor.
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 −3and want to rewrite it as the product
(k −3)(k + 1). That is,
k2−2k −3= (k −3)(k + 1).
Factoring
Recall from Section 6.1that this process is called
factoring the polynomial. Factoring reverses or
“undoes”multiplying.
Objective 1
Factor trinomials with a coefficient of 1 for the squared term. (cont’d)
Guidelines for factoring a trinomial of the form x2+ bx+
c are summarized here.
Find two integers whose product is c and whose sum is b.
1.Both integers must be positive if band care 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.
EXAMPLE 1
Factoring a Trinomial with All Positive Terms
Factor y2+ 12y + 20.
Solution:
| Factors of 20 | Sums of Factors |
| 1, 20 | 1 + 20 = 21 |
| 2, 10 | 2 + 10 = 12 |
| 4, 5 | 4 + 5 = 9You |
= (y + 10)( y + 2)
You can check your factoring by graphing both the unfactoredand factored forms of polynomials on your graphing calculators.
EXAMPLE 2
Factoring a Trinomial with a Negative Middle Term
Factor y2−10y+ 24.
Solution:
| 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 |
=(y - 6)( y - 4)
EXAMPLE 3
Factoring a Trinomial with Two Negative Terms
Factor a2−9a −22.
Solution:
| 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 |
=(a - 11)(a + 2)
EXAMPLE 4
Deciding whether Polynomials Are PrimeSlide
Factor if possible.
Solution:
| m2 - 8m + 14 | Factors of 14 | Sums of Factors | Prime |
| −1 , −14 | −1 + (−14) = −15 | ||
| −2 , −7 | −2 + (−7) = −9 | ||
| y2 + y + 2 | Factors of 2 | Sums of Factors | Prime |
| 1, 2 | 1 + 2 = 3 |
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.
EXAMPLE 5
Factoring a Trinomial with Two VariablesSlide
Factor r2 −6rs + 8s2.
Solution:
| Factors of 8s2 | Sums of Factors |
| −1s , −8s | −1s+ (−8s) = −9s |
| −2s, −4s | −2s+ (−4s) = −6s |
=(r - 4s)(r - 2s)
Objective 2
Factor such trinomials after
factoring out the greatest common
factor.
EXAMPLE 6
Factoring a Trinomial with a Common Factor
Factor 3x4−15x3+ 18x2.
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.