Factoring Polynomials

The Greatest Common Factor

Martin-Gay, Developmental Mathematics 3 Factors Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. Factoring – writing a polynomial as a product of polynomials.

Martin-Gay, Developmental Mathematics 4 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 1) Prime factor the numbers. 2) Identify common prime factors. 3) Take the product of all common prime factors. If there are no common prime factors, GCF is 1. Greatest Common Factor

Martin-Gay, Developmental Mathematics 5 1) x 3 and x 7 x 3 = x · x · x x 7 = x · x · x · x · x · x · x So the GCF is x · x · x = x 3 2) 6x 5 and 4x 3 6x 5 = 2 · 3 · x · x · x 4x 3 = 2 · 2 · x · x · x So the GCF is 2 · x · x · x = 2x 3 Find the GCF of each list of terms. Greatest Common Factor Example

Martin-Gay, Developmental Mathematics 6 Find the GCF of the following list of terms. a 3 b 2, a 2 b 5 and a 4 b 7 a 3 b 2 = a · a · a · b · b a 2 b 5 = a · a · b · b · b · b · b a 4 b 7 = a · a · a · a · b · b · b · b · b · b · b So the GCF is a · a · b · b = a 2 b 2 Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable. Greatest Common Factor Example

Martin-Gay, Developmental Mathematics 7 1) 6x 3 – 9x x = 3 · x · 2 · x 2 – 3 · x · 3 · x + 3 · x · 4 = 3x(2x 2 – 3x + 4) Factoring out the GCF Example

Martin-Gay, Developmental Mathematics 8 2) 14x 3 y + 7x 2 y – 7xy = 7 · x · y · 2 · x · x · y · x – 7 · x · y · 1 = 7xy(2x 2 + x – 1) Factoring out the GCF Example

Factoring Trinomials of the Form x 2 + bx + c by Grouping

Martin-Gay, Developmental Mathematics 10 Factoring a Four-Term Polynomial by Grouping 1)Arrange the terms so that the first two terms have a common factor and the last two terms have a common factor. 2)For each pair of terms, use the distributive property to factor out the pair’s greatest common factor. 3)If there is now a common binomial factor, factor it out. 4)If there is no common binomial factor in step 3, begin again, rearranging the terms differently. If no rearrangement leads to a common binomial factor, the polynomial cannot be factored. Remember: all have a common factor of 1! Factoring by Grouping

Martin-Gay, Developmental Mathematics 11 Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Factor xy + y + 2x + 2 by grouping. Notice that, although 1 is the GCF for all four terms of the polynomial, the first 2 terms have a GCF of y and the last 2 terms have a GCF of 2. xy + y + 2x + 2 y(x + 1) + 2(x + 1) = (x + 1)(y + 2) Factoring by Grouping Example

Martin-Gay, Developmental Mathematics 12 2)x 3 + 4x + x = x · x 2 + x · · x · 4 = x(x 2 + 4) + 1(x 2 + 4) = (x 2 + 4)(x + 1) Factor the following polynomial by grouping. Factoring by Grouping Example

Martin-Gay, Developmental Mathematics 13 3)2x 3 – x 2 – 10x + 5 = x 2 · 2x – x 2 · 1 – 5 · 2x – 5 · (– 1) = x 2 (2x – 1) – 5(2x – 1) = (2x – 1)(x 2 – 5) Factor the following polynomial by grouping. Factoring by Grouping Example

Martin-Gay, Developmental Mathematics 14 Factor 2x – 9y + 18 – xy by grouping. Neither pair has a common factor (other than 1). So, rearrange the order of the factors. 2x + 18 – 9y – xy = 2 · x + 2 · 9 – 9 · y – x · y = 2(x + 9) – y(9 + x) = 2(x + 9) – y(x + 9) = (make sure the factors are identical) (x + 9)(2 – y) Factoring by Grouping Example

Martin-Gay, Developmental Mathematics 15 Lab  Textbook page: