Copyright © 2011 Pearson Education, Inc. Factoring Polynomials Section P.5 Prerequisites

P.5 Copyright © 2011 Pearson Education, Inc. Slide P-3 Factoring “reverses” multiplication. By factoring, we can express a complicated polynomial as a product of several simpler expressions, which are often easier to study. When a monomial can be divided evenly into each term, we can use the distributive property to write abx 2 – ax = ax(bx – 1). We call this process factoring out ax. Both ax and bx – 1 are factors of abx 2 – ax. Since a is a factor of abx 2 and ax, a is a common factor of the terms of the polynomial. The greatest common factor (GCF) is a monomial that includes every number and variable that is a factor of all terms of the polynomial. The monomial ax is the greatest common factor of abx 2 – ax. Factoring Out the Greatest Common Factor

P.5 Copyright © 2011 Pearson Education, Inc. Slide P-4 Some four-term polynomials can be factored by grouping the terms in pairs and factoring out a common factor from each pair of terms. The trinomial that results from squaring a sum or a difference of two terms is called a perfect square trinomial. Factoring by Grouping

P.5 Copyright © 2011 Pearson Education, Inc. Slide P-5 To factor ax 2 + bx + c with a ≠ 1: 1. Find two numbers whose sum is b and whose product is ac. 2. Replace b by the sum of these two numbers. 3. Factor the resulting four-term polynomial by grouping. The ac-Method for Factoring

P.5 Copyright © 2011 Pearson Education, Inc. Slide P-6 Factoring the Special Products a 2 – b 2 = (a + b)(a – b)Difference of two squares a 2 + 2ab + b 2 = (a + b) 2 Perfect square trinomial a 2 – 2ab + b 2 = (a – b) 2 Perfect square trinomial Factoring the Difference and Sum of Two Cubes a 3 – b 3 = (a – b)(a 2 + ab + b 2 )Difference of two cubes a 3 + b 3 = (a + b)(a 2 – ab + b 2 )Sum of two cubes Factoring the Special Products

P.5 Copyright © 2011 Pearson Education, Inc. Slide P-7 When a polynomial involves a complicated expression, we can use two substitutions to help us factor. 1. First we replace the complicated expression by a single variable and factor the simpler-looking polynomial. 2. Then we replace the single variable by the complicated expression. This method is called substitution. A polynomial is factored when it is written as a product. Polynomials that cannot be factored using integral coefficients are called prime or irreducible over the integers. A polynomial is said to be factored completely when it is written as a product of prime polynomials. Factoring by Substitution