Factoring Polynomials By Dr. Carol A. Marinas © Copyright 2010 Carol A. Marinas

Factoring Polynomials Greatest Common Factor (GCF) Difference of Squares Perfect Square Trinomial General Trinomials with a = 1 Perfect Cubes Four-term Polynomials General Trinomials with a ≠ 1 Multiple Factoring Methods © Copyright 2010 Carol A. Marinas

Greatest Common Factor (GCF) Remove GCF first Example: ax 2 – 3a = a(x 2 – 3) Example 1: 24c – 12d = 12(2c – d) Example 2: 3x 2 – 6x – 12 = 3(x 2 – 2x – 4) © Copyright 2010 Carol A. Marinas

Difference of Perfect Squares Subtraction of 2 perfect squares Example: a 2 – b 2 = (a + b) (a – b) Example 1: 4c 2 – 9 = (2c + 3) (2c – 3) Example 2: 49x 6 y 4 – 81d 2 = (7x 3 y 2 + 9d) (7x 3 y 2 – 9d) © Copyright 2010 Carol A. Marinas

Perfect Square Trinomial First and last terms are perfect squares Middle term is twice the square root of the product of the first and last term Example: a 2 + 2ab + b 2 =(a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 Example 1: 4d 2 – 12d + 9 = (2d – 3) 2 Example 2: 16g 4 – 8g = (4g 2 – 1) 2 = [(2g + 1) (2g – 1)] 2 = (2g + 1) 2 (2g – 1) 2 © Copyright 2010 Carol A. Marinas

General Trinomials with a = 1 In the form: x 2 + bx + c Look for 2 factors of “c” that also add to “b” Example 1: x 2 + 5x + 6 = (x + 3) (x + 2) Example 2: x 2 – x – 42 = (x – 7) (x + 6) © Copyright 2010 Carol A. Marinas

Perfect Cubes Sum or Difference of 2 perfect cubes Examples: a 3 x 3 – b 3 = (ax – b) (a 2 x 2 + abx + b 2 ) a 3 x 3 + b 3 = (ax + b) (a 2 x 2 – abx + b 2 ) Same Opposite Always Plus Example 1: 27x 3 – 8 = (3x – 2) (9x 2 + 6x + 4) Example 2: 125b = (5b + 1) (25b 2 – 5b + 1) © Copyright 2010 Carol A. Marinas

Four-term Polynomials Factor by Grouping Example: ax 2 + 2ay + 3x 2 + 6y = a(x 2 + 2y) + 3(x 2 +2y) = (a + 3) (x 2 + 2y) Example 1: 7ax ag + 5x g = 7a(x 2 + 2g) + 5(x 2 + 2g) = (7a + 5) (x 2 + 2g) Example 2: 6cx 2 – 5cx – 12x + 10 = cx (6x – 5) – 2(6x – 5) = (cx – 2) (6x – 5) © Copyright 2010 Carol A. Marinas

General Trinomial with a ≠ 1 Example 1: 12c 2 – 16c – 3 = 12c 2 – 18c + 2c – 3 = 6c(2c – 3) + 1(2c – 3) = (6c + 1) (2c – 3) Example 2: 2x 2 + 7x – 15 = 2x x – 3x – 15 = 2x(x + 5) – 3(x + 5) = (2x – 3)(x + 5) © Copyright 2010 Carol A. Marinas

Multiple Factoring Methods Factor out GCF first Then go to other methods Example 1: 3x 2 – 3y 2 = 3(x 2 – y 2 ) = 3(x + y) (x – y) Example 2: 16x 3 – 54 = 2(8x 3 – 27) = 2(2x – 3) (4x 2 + 6x + 9) Example 3: 2x x + 18 = 2(x 2 + 6x + 9) = 2(x + 3) 2 © Copyright 2010 Carol A. Marinas