Factoring Polynomials ARC INSTRUCTIONAL VIDEO MAT 120 COLLEGE ALGEBRA

Factors  When an integer is written as a product of integers, each of the integers in the product is a factor of the original number  Factoring – writing a polynomial as a product of polynomials.

Greatest Common Factor  Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved.  Finding the GCF of a list of integers or terms  Prime factor the numbers  Identify common prime factors  Take the product of all common prime factors  If there are no common prime factors, GCF is 1

Greatest Common Factor  Example:  Find the GCF of each list of numbers  6x 5 and 4x 3  6x 5 = 2 · 3 · x · x · x · x · x  4x 3 = 2 · 2 · x · x · x  So the GCF is 2 · x · x · x = 2x 3  12 and 8  12 = 2 · 2 · 3  8 = 2 · 2 · 2  So the GCF is 2 · 2 = 4.

Factoring Polynomials  The first step in factoring a polynomial is to find the GCF of all its terms  Then we write the polynomial as a product by factoring out the GCF from all the terms  The remaining factors in each term will form a polynomial

Factoring out the GCF  Example:  Factor out the GCF in each of the following polynomials  6x 3 – 9x x =  3 · x · 2 · x 2 – 3 · x · 3 · x + 3 · x · 4 =  3x(2x 2 – 3x + 4)  6(x + 2) – y(x + 2) =  6 · (x + 2) – y · (x + 2) =  (x + 2)(6 – y)

Factoring  Remember that factoring out the GCF from the terms of a polynomial should always be the first step in factoring a polynomial.  This will usually be followed by additional steps in the process  Example:  Factor y 2 – 18x – 3xy y 2 – 18x – 3xy 2 = 3(30 + 5y 2 – 6x – xy 2 ) = 3(5 · · y 2 – 6 · x – x · y 2 ) = 3(5(6 + y 2 ) – x (6 + y 2 )) = 3(6 + y 2 )(5 – x)