Factoring and Solving Equations Copyright © Cengage Learning. All rights reserved.

6.1 Factoring Polynomials with Common Factors

What You Will Learn Find the greatest common factor of two or more expressions Factor out the greatest common monomials factor from polynomials Factor polynomials by grouping

Greatest Common Factor

Greatest Common Factor You have used the Distributive Property to multiply polynomials. In this chapter, you will study the reverse process, which is factoring. Multiplying Polynomials Factoring Polynomials

Greatest Common Factor To factor an expression efficiently, you need to understand the concept of the greatest common factor of two (or more) integers or terms. You have learned that the greatest common factor of two or more integers is the greatest integer that is a factor of each integer. For example, the greatest common factor of 12 = 2  2  3 and 30 = 2  3  5 is 2  3 = 6.

Example 1 – Finding the Greatest Common Factor To find the greatest common factor of 5x2y2 and 30x3y, first factor each term. 5x2y2 = 5  x  x  y  y = (5x2y)(y) 30x3y = 2  3  5  x  x  x  y = (5x2y)(6x) So, you can conclude that the greatest common factor is 5x2y.

Example 2 – Finding the Greatest Common Factor To find the greatest common factor of 8x5, 20x3, and 16x4 first factor each term. 8x5 = 2  2  2  x  x  x  x  x = (4x3)(4x2) 20x3 = 2  2  5  x  x  x = (4x3)(5) 16x4 = 2  2  2  2  x  x  x  x = (4x3)(4x) So, you can conclude that the greatest common factor is 4x3.

Common Monomial Factor

Dividing a Polynomial by a Binomial Consider the polynomial 8x5 + 16x4 + 20x3. The greatest common factor, 4x3, of these terms is the greatest common monomial factor of the polynomial. When you use the Distributive Property to remove this factor from each term of the polynomial, you are factoring out the greatest common monomial factor. 8x5 + 16x4 + 20x3 = 4x3(2x2) + 4x3(4x) + 4x3(5) = 4x3(2x2 + 4x + 5) Factor each term. Factor out common monomial factor.

Example 3 – Greatest Common Monomial Factor Factor out the greatest common monomial factor from 6x – 18. Solution: The greatest common integer factor of 6x and 18 is 6. There is no common variable factor. 6x – 18 = 6(x) – 6(3) = 6(x – 3) Greatest common monomial factor is 6. Factor 6 out of each term.

Example 4 – Greatest Common Monomial Factor Factor out the greatest common monomial factor from 10y3 – 25y2. Solution: For the terms 10y3 – 25y2, 5 is the greatest common integers factor and y2 is the highest-power common variable factor. 10y3 – 25y2 = 5y2(2y) – 5y2(5) = 5y2(2y – 5) Greatest common factor is 5y2. Factor 5y2 out of each term.

Greatest Common Monomial The greatest common monomial factor of the terms of a polynomial is usually considered to have a positive coefficient. However, sometimes it is convenient to factor a negative number out of a polynomial.

Example 8 – A Negative Common Monomial Factor Factor the polynomial –2x2 + 8x – 12 in two ways. a. Factor out a common monomial factor of 2. b. Factor out a common monomial factor of –2. Solution: a. To factor out the common monomial factor of 2, write the following. –2x2 + 8x – 12 = 2(–x2) + 2(4x) + 2(–6) = 2(–x2 + 4x – 6) Factor each term. Factored form

Example 8 – A Negative Common Monomial Factor cont’d b. To factor –2 out of the polynomial, write the following. –2x2 + 8x – 12 = –2(x2) + (–2)(–4x) + (–2)(6) = –2(x2 – 4x + 6) Check this result by multiplying (x2 – 4x + 6) by –2. When you do, you will obtain the original polynomial. Factor each term. Factored form

Greatest Common Monomial With experience, you should be able to omit writing the first step shown in Example 8. For instance, to factor –2 out of –2x2 + 8x – 12, you could simply write –2x2 + 8x – 12 = –2(x2 – 4x + 6).

Worksheet

Factoring by Grouping

Factoring by Grouping There are occasions when the common factor of an expression is not simply a monomial. For instance, the expression x2(x – 2) + 3(x – 2) has the common binomial factor (x – 2). Factoring out this common factor produces x2(x – 2) + 3(x – 2) = (x – 2)(x2 + 3). This type of factoring is part of a more general procedure called factoring by grouping.

Example 9 – Common Binomial Factors Factor each expression. a. 5x2(7x – 1) – 3(7x – 1) b. 2x(3x – 4) + (3x – 4) c. 3y2(y – 3) + 4(3 – y) Solution: a. Each of the terms of this expression has a binomial factor of (7x – 1). 5x2(7x – 1) – 3(7x – 1) = (7x – 1)(5x2 – 3)

Example 9 – Common Binomial Factors cont’d b. Each of the terms of this expression has a binomial factor of (3x – 4). 2x(3x – 4) + (3x – 4) = (3x – 4)(2x + 1) Be sure you see that when (3x – 4) is factored out of itself, you are left with the factor 1. This follows from the fact that (3x – 4)(1) = (3x – 4). c. 3y2(y – 3) + 4(3 – y) = 3y2(y – 3) – 4(y – 3) = (y – 3)(3y2 – 4) Write 4(3 – y) as –4(y – 3). Common factor is (y – 3).

Factoring by Grouping In Example 9, the polynomials were already grouped so that it was easy to determine the common binomial factors. In practice, you will have to do the grouping as well as the factoring. To see how this works, consider the expression x3 + 2x2 + 3x + 6 and try to factor it. Note first that there is no common monomial factor to take out of all four terms.

Factoring by Grouping But suppose you group the first two terms together and the last two terms together. x3 + 2x2 + 3x + 6 = (x3 + 2x2) + (3x + 6) = x2(x + 2) + 3(x + 2) = (x + 2)(x2 + 3) When factoring by grouping, be sure to group terms that have a common monomial factor. For example, in the polynomial above, you should not group the first term x3 with the fourth term 6. Group terms. Factor out common monomial factor in each group. Factored form

Example 10 – Factoring by Grouping Factor x3 + 2x2 + x + 2. Solution: x3 + 2x2 + x + 2 = (x3 + 2x2) + (x + 2) = x2(x + 2) + (x + 2) = (x + 2)(x2 + 1) Group terms. Factor out common monomial factor in each group. Factored form

Factoring by Grouping Note that in Example 10 the polynomial is factored by grouping the first and second terms and the third and fourth terms. You could just as easily have grouped the first and third terms and the second and fourth terms, as follows. x3 + 2x2 + x + 2 = (x3 + x) + (2x2 + 2) = x(x2 + 1) + 2(x2 + 1) = (x2 + 1)(x + 2) You can always check to see that you have factored an expression correctly by multiplying the factors and comparing the result with the original expression.

Example 12 – Geometry: Area of a Rectangle The area of a rectangle of width (2x – 1) feet is (2x3 + 2x – x2) square feet, as shown below. Factor this expression to determine the length of the rectangle.

Example 12 – Geometry: Area of a Rectangle cont’d Solution Verbal Model: Labels: Area = 2x3 + 2x – x2 (square feet) Width = 2x – 1 (feet) Equation: 2x3 + 4x – x2 – 2 = (2x3 + 4x) + (–x2 – 2) = 2x(x2 + 2) + (x2 + 2) = (x2 + 2)(2x – 1) The length of the rectangle is (x2 + 2) feet. Group terms. Factor out common monomial factor in each group. Factored form

Factor the Polynomials