Factoring Perfect-Square Trinomials and Differences of Squares 5.5 Factoring Perfect-Square Trinomials and Differences of Squares Perfect-Square Trinomials Differences of Squares More Factoring by Grouping Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Perfect-Square Trinomials Consider the trinomial x2 +8x + 16 To factor it, we can look for factors of 16 that add to 8. These factors are 4 and 4 and the factorization is x2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)2 Note that the result is the square of a binomial. x2 + 8x + 16 is called a perfect-square trinomial. A trinomial, recognized as a perfect-square trinomial can be quickly factored. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Recognize a Perfect-Square Trinomial Two terms must be squares, such as A2 and B2. There must be no minus sign before A2 or B2. The remaining term must be 2AB or its opposite, –2AB. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Perfect-Square Trinomial A2 + 2AB + B2 = (A + B)2; A2 – 2AB + B2 = (A – B)2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor m2 + 10m + 25. Solution Using the formula: A2 + 2AB + B2 = (A + B)2 we have: m2+ 10m + 25 = m2 + 2(m)(5) + 52 = (m + 5) 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor 5x2 – 10xy + 5y2. Solution 5x2 – 10xy + 5y2 = 5(x2 – 2xy + y2) = 5(x2 – 2(x)(y) + y2) = 5(x – y) 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Differences of Squares When an expression like x2 – 49 is recognized as a difference of two squares, we can reverse another pattern first seen in Section 5.2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Factoring a Difference of Two Squares A2 – B2 = (A + B)(A – B) To factor a difference of two squares, write the product of the sum and difference of the quantities being squared. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor x2 – 49. Solution A2 – B2 = (A + B)(A – B) x 2 – 7 2 = (x + 7)(x – 7) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor 3m2n4 – 27n2. Solution 3m2n4 – 27n2 = 3n2(m2n2 – 9) Factor out 3n2 = 3n2(mn + 3)(mn – 3) Factor the difference of two squares Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More Factoring by Grouping Sometimes, when factoring a polynomial with four terms, we may be able to factor further. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Factor: y6 – 9y4 + 2y2 – 18. Solution y6 – 9y4 + 2y2 – 18 Factor out y2 – 9 y2 – 9 can be factored further = (y4 + 2)(y – 3)(y + 3) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley