Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 4.4 Factoring Trinomials of the Form ax 2 + bx + c by Grouping

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. An alternative method that can be used to factor trinomials of the form ax 2 + bx + c, a ≠ 1 is called the grouping method. This method uses factoring by grouping. Factor xy + y + 2x + 2 by grouping. Notice that, although 1 is the GCF for all four terms of the polynomial, the first 2 terms have a GCF of y and the last 2 terms have a GCF of 2. xy + y + 2x + 2 = x · y + 1 · y + 2 · x + 2 · 1 = y(x + 1) + 2(x + 1) = (x + 1)(y + 2) Factoring by Grouping Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Factor Trinomials by Grouping Step 1:Factor out a greatest common factor, if there is one other than 1. Step 2:For the resulting trinomial ax 2 + bx + c, find two numbers whose product is a c and whose sum is b. Step 3:Write the middle term, bx, using the factors found in Step 2. Step 4:Factor by grouping. Factoring by Grouping

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 1) x 3 + 4x + x = x · x 2 + x · · x · 4 = x(x 2 + 4) + 1(x 2 + 4) = (x 2 + 4)(x + 1) 2)2x 3 – x 2 – 10x + 5 = x 2 · 2x – x 2 · 1 – 5 · 2x – 5 · (– 1) = x 2 (2x – 1) – 5(2x – 1) = (2x – 1)(x 2 – 5) Factor each of the following polynomials by grouping. Factoring by Grouping Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor 2x – 9y + 18 – xy by grouping. Neither pair has a common factor (other than 1). So, rearrange the order of the factors. 2x + 18 – 9y – xy = 2 · x + 2 · 9 – 9 · y – x · y = 2(x + 9) – y(9 + x) = 2(x + 9) – y(x + 9) Make sure the factors are identical. = (x + 9)(2 – y) Factoring by Grouping Example