Chapter 6 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More on Factoring Trinomials Factor trinomials by grouping when the coefficient of the squared term is not 1. Factor trinomials by using the FOIL method

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trinomials such as 2x 2 + 7x + 6, in which the coefficient of the squared term is not 1, are factored with extensions of the methods from the previous sections. One such method uses factoring by grouping from Section 6.1. More on Factoring Trinomials Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Factor trinomials by grouping when the coefficient of the squared term is not 1. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor trinomials by grouping when the coefficient of the squared term is not 1. Recall that a trinomial such as m 2 + 3m + 2 is factored by finding two numbers whose product is 2 and whose sum is 3. To factor 2x 2 + 7x + 6, we look for two integers whose product is 2 · 6 = 12 and whose sum is 7. Slide Sum Product is 2 · 6 = 12

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor trinomials by grouping when the coefficient of the squared term is not 1. (cont’d) Slide By considering pairs of positive integers whose product is 12, we find the necessary integers to be 3 and 4. We use these integers to write the middle term, 7x, as 7x = 3x + 4x. The trinomial 2x 2 + 7x + 6 becomes

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Factor. Factoring Trinomials by Grouping Slide Solution :

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Factor 6p p 3 + 9p 2. Solution: Factoring a Trinomial with a Common Factor by Grouping Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Factor trinomials by using the FOIL method. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To factor 2x 2 + 7x + 6, again using an alternate method explained in Section 6.2, we use the FOIL method in reverse. We want to write the equation 2x 2 + 7x + 6 as the product of two binomials. Factor trinomials by using the FOIL method. Slide The product of the two first terms of the binomials is 2x 2. The possible factors of 2x 2 are 2x and x or −2x and −x. Since all terms of the trinomial are positive, we consider only positive factors. Thus, we have

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The product of the two last terms, 6, can be factored as 1 · 6, 6 · 1, 3 · 2, or 3 · 2. Try each pair to find the pair that gives the correct middle term, 7x. Factor trinomials by using the FOIL method. (cont’d) Slide If the terms of the original polynomial have greatest common factor 1, then all of that polynomials binomial factors also have GCF 1. Incorrect Now try the number 2 and 3 as factors of 6. Because of the common factor 2 in 2x + 2, (2x + 2)(x + 3) will not work, so we try (2x + 3)(x + 2). Correct

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Factoring a Trinomial with All Positive Terms by Using FOIL Slide Factor 6p p Incorrect Correct

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Factoring a Trinomial with a Negative Middle Term by Using FOIL Slide Factor 10m 2 – 23m Incorrect Correct

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Factor 5p p – 6. Solution: Factoring a Trinomial with a Negative Last Term by Using FOIL Slide Correct Incorrect

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor 6m mn – 10n 2. EXAMPLE 6 Factoring a Trinomial with Two Variables Slide Solution: CorrectIncorrect

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Factoring Trinomials with Common Factors Slide Factor. Solution: