Copyright © 2016, 2012, 2008 Pearson Education, Inc. 1 Factoring Trinomials with the leading coefficient of 1

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 2 Factoring Trinomials 1. Factor trinomials with a coefficient of 1 for the second-degree term. 2. Factor such trinomials after factoring out the greatest common factor.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 3 Factoring Trinomials Using the FOIL method, we see that the product of the binomial k − 3 and k +1 is Suppose instead that we are given the polynomial k 2 – 2k – 3 and want to write it as the product (k – 3)(k + 1). Recall that factoring is a process that reverses, or “undoes,” multiplying.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 4 Objective 1 Factor trinomials with a coefficient of 1 for the second-degree term.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 5 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 x 2 + 5x + 6 by finding integers m and n such that x 2 + 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:

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 6 Comparing this result with x 2 + 5x + 6 shows that we must find integers m and n having a sum of 5 and a product of 6. Factor trinomials with a coefficient of 1 for the second-degree term.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 7 Because 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 we consider only pairs in which both integers are positive. Both pairs have a product of 6, but only the pair 3 and 2 has a sum of 5. So 3 and 2 are the required integers. Factor trinomials with a coefficient of 1 for the second-degree term.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 8 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. Factor trinomials with a coefficient of 1 for the second-degree term.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 9 Factor x 2 + 9x Look for two integers whose product is 18 and whose sum is 9. List pairs of integers whose product is 18, and examine the sums. Example 1 Factors of 18Sum of Factors Sum is 9. 1, 18 2, 9 3, 6 Factoring a Trinomial (All Positive Terms)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 10 Factor. Look for two integers whose product is 24 and whose sum is –10. Example 2 Factors of 24Sum of Factors Sum is –10. –1, –24 –2, –12 –3, –8 –4, –6 Factoring a Trinomial (Negative Middle Term)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 11 Factor. Look for two integers whose product is –30 and whose sum is 1. Example 3 Factors of –30Sum of Factors Sum is 1. –1, 30 1, – 30 5, – 6 –5, 6 Factoring a Trinomial (Negative Last [Constant] Term)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 12 Factor. Look for two integers whose product is –22 and whose sum is –9. Example 4 Factors of –22Sum of Factors Sum is –9. –1, 22 –1, – 22 –2, 11 –2, –11 Factoring a Trinomial (Two Negative Terms)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 13 Prime Polynomials Trinomials that cannot be factored using only integers are prime polynomials.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 14 Factor each trinomial if possible. a. m 2 – 8m + 14 None of the pairs of integers has a sum of –8. Therefore, the trinomial cannot be factored using only integers. It is a prime polynomial. Example 5 Factors of 14Sum of Factors –1, –14–1 + (–14) = –15 –2, –7–2 + (–7) = –9 Deciding Whether Polynomials Are Prime

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 15 Factor each trinomial if possible. b. y 2 + y + 2 None of the pairs of integers has a sum of 1. Therefore, the trinomial cannot be factored using only integers. It is a prime polynomial. Example 5 Factors of 2Sum of Factors 1, = 3 –1, –2–1 + (–2) = –3 Deciding Whether Polynomials Are Prime (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 16 Guidelines for Factoring x 2 + 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. (See Example 1.) 2. Both integers must be negative if c is positive and b is negative. (See Example 2.) 3. One integer must be positive and one must be negative if c is negative. (See Examples 3 and 4.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 17 Factor Example 6 Factors ofSum of Factors –1s, –8s –2s, –4s Factoring a Multivariable Trinomial

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 18 Objective 2 Factor such trinomials after factoring out the greatest common factor.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 19 Factor Factor out the GCF, 3x 2. Factor x 2 – 5x + 6. The integers –3 and –2 have a product of 6 and a sum of –5. Example 7 Factoring a Trinomial with a Common Factor