Polynomials: Factoring - XYZ Custom Plus

742Chapter 12 Polynomials: Factoring3. Factor the greatest commonfactor from:5x + 15Example 3Factor the greatest common factor from 3x − 15.Solution The greatest common factor for the terms 3x and 15 is 3. We canrewrite both 3x and 15 so that the greatest common factor 3 is showing in eachterm. It is important to realize that 3x means 3 ⋅ x. The 3 and the x are not “stuck”together:3x − 15 = 3 ⋅ x − 3 ⋅ 5Now, applying the distributive property, we have:3 ⋅ x − 3 ⋅ 5 = 3(x − 5)To check a factoring problem like this, we can multiply 3 and x − 5 to get 3x − 15,which is what we started with. Factoring is simply a procedure by which wechange sums and differences into products. In this case we changed the difference3x − 15 into the product 3(x − 5). Note, however, that we have not changedthe meaning or value of the expression. The expression we end up with is equivalentto the expression we started with.4. Factor the greatest commonfactor from:25x 4 − 35x 3Example 4Factor the greatest common factor from:5x 3 − 15x 2SolutionThe greatest common factor is 5x 2 . We rewrite the polynomial as:5x 3 − 15x 2 = 5x 2 ⋅ x − 5x 2 ⋅ 3Then we apply the distributive property to get:5x 2 ⋅ x − 5x 2 ⋅ 3 = 5x 2 (x − 3)To check our work, we simply multiply 5x 2 and (x − 3) to get 5x 3 − 15x 2 , which isour original polynomial.5. Factor the greatest commonfactor from:20x 8 − 12x 7 + 16x 6Example 5Factor the greatest common factor from:16x 5 − 20x 4 + 8x 3Solution The greatest common factor is 4x 3 . We rewrite the polynomial so wecan see the greatest common factor 4x 3 in each term; then we apply the distributiveproperty to factor it out.1 6 x 5 − 20x 4 + 8x 3 = 4x 3 ⋅ 4x 2 − 4x 3 ⋅ 5x + 4x 3 ⋅ 2= 4x 3 (4x 2 − 5x + 2)6. Factor the greatest commonfactor from:8xy 3 − 16x 2 y 2 + 8x 3 yExample 6Factor the greatest common factor from:6x 3 y − 18x 2 y 2 + 12xy 3Solution The greatest common factor is 6xy. We rewrite the polynomial interms of 6xy and then apply the distributive property as follows:6x 3 y − 18x 2 y 2 + 12xy 3 = 6xy ⋅ x 2 − 6xy ⋅ 3xy + 6xy ⋅ 2y 2= 6xy(x 2 − 3xy + 2y 2 )Answers3. 5(x + 3) 4. 5x 3 (5x − 7)5. 4x 6 (5x 2 − 3x + 4)6. 8xy( y 2 − 2xy + x 2 )