Chapter 6 Section 2

6.2 Factoring Trinomials Factor trinomials with a coefficient of 1 for the second-degree term. Factor such trinomials after factoring out the greatest common factor. 2

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

Factor trinomials with a coefficient of 1 for the second-degree term. Objective 1 Factor trinomials with a coefficient of 1 for the second-degree term. Slide 6.2-4

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). To find these integers m and n, we first use FOIL to multiply the two binomials on the right side of the equation: 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. Sum of m and n is 5. Product of m and n is 6. Slide 6.2-5

Factor trinomials with a coefficient of 1 for the second-degree term Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d) 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

Factors of 20 Sums of Factors 1, 20 1 + 20 = 21 2, 10 2 + 10 = 12 4, 5 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 = 9 You can check your factoring by graphing both the unfactored and factored forms of polynomials on your graphing calculators. Slide 6.2-7

Factors of 24 Sums of Factors − 1 , −24 −1 + (−24) = −25 −2 , −12 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 Slide 6.2-8

Factors of − 30 Sums of Factors − 1 , 30 −1 + (30) = 29 1 , − 30 EXAMPLE 3 Factoring a Trinomial with a Negative Last (Constant) Term Factor z2 + z − 30. Solution: 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

Factors of −22 Sums of Factors −1 , 22 −1 + 22 = 21 1, −22 EXAMPLE 4 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 Slide 6.2-10

Factor trinomials with a coefficient of 1 for the second-degree term Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d) Some trinomials cannot be factored by using only integers. We call such trinomials prime polynomials. Summarize the signs of the binomials when factoring a trinomial whose leading coefficient is positive. If the last term of the trinomial is positive, both binomials will have the same “middle” sign as the second term. If the last term of the trinomial is negative, the binomials will have one plus and one minus “middle” sign. Slide 6.2-11

−1 , −14 −1 + (−14) = −15 −2 , −7 −2 + (−7) = −9 Factors of 14 EXAMPLE 5 Deciding Whether Polynomials Are Prime Solution: Factor if possible. Factors of 14 Sums of Factors −1 , −14 −1 + (−14) = −15 −2 , −7 −2 + (−7) = −9 Prime Factors of 2 Sums of Factors 1, 2 1 + 2 = 3 Prime Slide 6.2-12

Guidelines for Factoring x2 + bx + c Factor trinomials with a coefficient of 1 for the second-degree term. (cont’d) 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

Factors of 8s2 Sums of Factors −1s , −8s − 1s + (−8s) = −9s −2s , −4s EXAMPLE 6 Factoring a Trinomial with Two Variables Factor r2 − 6rs + 8s2. Solution: Factors of 8s2 Sums of Factors −1s , −8s − 1s + (−8s) = −9s −2s , −4s −2s + (−4s) = −6s Slide 6.2-14

Factor such trinomials after factoring out the greatest common factor. Objective 2 Factor such trinomials after factoring out the greatest common factor. Slide 6.2-15

Factor such trinomials after factoring out the greatest common factor. If a trinomial has a common factor, first factor it out. Slide 6.2-16

Factoring a Trinomial with a Common Factor EXAMPLE 7 Factoring a Trinomial with a Common Factor Factor 3x4 − 15x3 + 18x2. 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

HOME WORK 6.2