Chapter 6 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 1 2 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 More on Factoring Trinomials Trinomials such as 2x2 + 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 3 Objective 1 Factor trinomials by grouping when the coefficient of the squared term is not 1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 4 Factor trinomials by grouping when the coefficient of the squared term is not 1. Recall that a trinomial such as m2 + 3m + 2 is factored by finding two numbers whose product is 2 and whose sum is 3. To factor 2x2 + 7x + 6, we look for two integers whose product is 2 · 6 = 12 and whose sum is 7. Sum 2 x2 7 x 6 Product is 2 · 6 = 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 5 Factor trinomials by grouping when the coefficient of the squared term is not 1. (cont’d) 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 2x2 + 7x + 6 becomes 2 x 2 7 x 6 2 x 2 3x 4 x 6. 2 x 2 3x 4 x 6 x 2x 3 2 2x 3 x 2 2x 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 6 EXAMPLE 1 Factoring Trinomials by Grouping Factor. Solution: 3 p2 4 p 1 3 p2 3 p 1 p 1 3 p 2 3 p 1 p 1 3 p p 1 1 p 1 3 p 1 p 1 12 z 16 z 3 12 z 18z 2 z 3 12 z 2 18 z 2 z 3 2 2 6z 2z 3 1(2z 3) 6z 1 2z 3 8r 2 6rt 5t 2 8r 2 10rt 4rt 5t 2 8r 2 10rt 4rt 5t 2 2r 4r 5t 1t 4r 5t 2r t 4r 5t Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 7 EXAMPLE 2 Factoring a Trinomial with a Common Factor by Grouping Factor 6p4 + 21p3 + 9p2. Solution: 3 p 2 2 p 2 7 p 3 3 p 2 p 6 p 1 p 3 2 2 2 3 p 2 p 6 p 1 p 3 3 p 2 2 p p 3 1 p 3 2 3 p2 2 p 1 p 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 8 Objective 2 Factor trinomials by using the FOIL method. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 9 Factor trinomials by using the FOIL method. To factor 2x2 + 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 2x2 + 7x + 6 as the product of two binomials. 2 x2 7 x 6 The product of the two first terms of the binomials is 2x2. The possible factors of 2x2 are 2x and x or −2x and −x. Since all terms of the trinomial are positive, we consider only positive factors. Thus, we have 2 x2 7 x 6 2 x x . Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 10 Factor trinomials by using the FOIL method. (cont’d) 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. 2x 1 x 6 2x 6 x 1 x 6x 12x 2x Incorrect Incorrect 13x 8x If the terms of the original polynomial have greatest common factor 1, then all of that polynomials binomial factors also have GCF 1. 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). 2x 3 x 2 3x 4x Correct 7x Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 11 EXAMPLE 3 Factoring a Trinomial with All Positive Terms by Using FOIL Factor 6p2 + 19p + 10. Solution: 6 p 21p 5 6 p 101p 1 2p 10 p 30 p 6p Incorrect Incorrect 32 p 16 p 2 p 103 p 1 3 p 2 2 p 5 13p 4p 2p 15 p Incorrect Correct 15 p 19 p Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 12 EXAMPLE 4 Factoring a Trinomial with a Negative Middle Term by Using FOIL Factor 10m2 – 23m + 12. Solution: 2m 125m 1 60m 2m 62m 10m 21m 6 Incorrect 2m 60m 62m Incorrect 2m 35m 4 Correct 15m 8m 23m Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 13 EXAMPLE 5 Factoring a Trinomial with a Negative Last Term by Using FOIL Factor 5p2 + 13p – 6. Solution: 5 p 3 p 2 Incorrect 3p 15p 12 p 5 p 2 p 3 2 p 15p 13p Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Correct Slide 6.3 - 14 EXAMPLE 6 Factoring a Trinomial with Two Variables Factor 6m2 + 11mn – 10n2. Solution: 6m 10n m 1n 3m 2n 2m 5n 4mn 15mn 11mn 10mn 6mn 4mn Incorrect Correct Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.3 - 15 EXAMPLE 7 Factor. 28x4 58x3 30 x2 Solution: 2 x 14 x 29 x 15 2 2 2x2 7x 3 2x 5 Factoring Trinomials with Common Factors 24x3 32x2 y 6xy 2 2 x 12 x 2 16 xy 3 y 2 2x 6x y 2x 3 y 6x 35x 29 x Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2xy 18xy 16xy Slide 6.3 - 16

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