1. © 2010 Pearson Prentice Hall. All rights reserved Removing Common Factors; Factoring by Grouping

2. Tobey & Slater, Intermediate Algebra, 6e 2 Greatest Common Factor When two or more numbers, variables, or algebraic expressions are multiplied, each is called a factor. 5 · 3 Factor 5x 2 · 3y 4 Factor The greatest common factor is the largest factor that is common to all terms of the expression. It contains the largest possible common factor of the numerical coefficients and the largest possible common variable factor. 12xy + 6xz = 6x(2y + z) 6x is the greatest common factor.

3. Tobey & Slater, Intermediate Algebra, 6e 3 Factoring with Common Factors Example: Factor. 36a 6 + 45a 4 – 18a 2 = 9a 2 (4a 4 + 5a 2 – 2) 9a 2 is the greatest common factor. Example: Factor. 6(3x + y) – z(3x + y) = (3x + y)(6 – z) (3x + y) is the common factor.

4. Tobey & Slater, Intermediate Algebra, 6e 4 Factoring by Grouping Example: Factor by grouping. x 2 + 7x + 3x + 21 x 2 + 7x + 3x + 21 x is the common factor.3 is the common factor. = x(x + 7) + 3(x + 7) (x + 7)(x + 3) = x 2 + 3x + 7x + 21 Check: = (x + 7) (x + 3) Factor out the 3 from the last two terms. Factor out the x from the first two terms. These two factors need to be the same.

5. Tobey & Slater, Intermediate Algebra, 6e 5 Factoring by Grouping Example: Factor by grouping. x 2 + 7x + 3x + 21 x 2 + 7x + 3x + 21 x is the common factor.3 is the common factor. x 2 + 7x + 3x + 21 = x(x + 7) (x + 3)(x + 7) = x 2 + 7x + 3x + 21 Check: These two factors need to be the same. + 3(x + 7) Factor out the x from the first two terms. Factor out the 3 from the last two terms. = (x + 3)(x + 7)

6. Tobey & Slater, Intermediate Algebra, 6e 6 Factoring by Grouping Example: Factor by grouping. xy – 4x – 3y + 12 xy – 4x – 3y + 12 x is the common factor.– 3 is the common factor. = x(y – 4) + (– 3)(y – 4) (A negative factor is needed to obtain the same expression in the parentheses.) = (y – 4)(x – 3) multiply to check! Factor out the x. Factor out the – 3. (Careful!)

7. Tobey & Slater, Intermediate Algebra, 6e 7 When to use factor by grouping 18x – 3xz + 6y – yz 1x 2 + 7x + 3x + 21 xy – 4x – 3y + 12 Look for common relationships between coefficients and variables in each pair of terms. (You may need to change the order of the terms) 1 x 7 = 73 x 7 = 21 x goes down 1 degree Review these:

8. Tobey & Slater, Intermediate Algebra, 6e 8 Try this. Factor out the GCF: a) b) Factor by grouping: c)6x 2 – 2x + 3xy – y d) 5y 2 + y – 10y 3 – 2y 2