6.1 Factoring Polynomials with Common Factors. Copyright Cengage Learning. All rights reserved.


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1 6.1 Factoring Polynomials with Common Factors Copyright Cengage Learning. All rights reserved. 1
2 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 2
3 Greatest Common Factor 3
4 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 4
5 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 = and 30 = is 2 3 = 6. 5
6 Example 1 Finding the Greatest Common Factor To find the greatest common factor of 5x 2 y 2 and 30x 3 y, first factor each term. 5x 2 y 2 = 5 x x y y = (5x 2 y)(y) 30x 3 y = x x x y = (5x 2 y)(6x) So, you can conclude that the greatest common factor is 5x 2 y. 6
7 Example 2 Finding the Greatest Common Factor To find the greatest common factor of 8x 5, 20x 3, and 16x 4 first factor each term. 8x 5 = x x x x x = (4x 3 )(4x 2 ) 20x 3 = x x x = (4x 3 )(5) 16x 4 = x x x x = (4x 3 )(4x) So, you can conclude that the greatest common factor is 4x 3. 7
8 Common Monomial Factor 8
9 Dividing a Polynomial by a Binomial Consider the polynomial 8x x x 3. The greatest common factor, 4x 3, 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. 8x x x 3 = 4x 3 (2x 2 ) + 4x 3 (4x) + 4x 3 (5) = 4x 3 (2x 2 + 4x + 5) Factor each term. Factor out common monomial factor. 9
10 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) Greatest common monomial factor is 6. = 6(x 3) Factor 6 out of each term. 10
11 Example 4 Greatest Common Monomial Factor Factor out the greatest common monomial factor from 10y 3 25y 2. Solution: For the terms 10y 3 25y 2, 5 is the greatest common integers factor and y 2 is the highestpower common variable factor. 10y 3 25y 2 = 5y 2 (2y) 5y 2 (5) = 5y 2 (2y 5) Greatest common factor is 5y 2. Factor 5y 2 out of each term. 11
12 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. 12
13 Example 8 A Negative Common Monomial Factor Factor the polynomial 2x 2 + 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. 2x 2 + 8x 12 = 2( x 2 ) + 2(4x) + 2( 6) Factor each term. = 2( x 2 + 4x 6) Factored form 13
14 Example 8 A Negative Common Monomial Factor b. To factor 2 out of the polynomial, write the following. 2x 2 + 8x 12 = 2(x 2 ) + ( 2)( 4x) + ( 2)(6) cont d Factor each term. = 2(x 2 4x + 6) Factored form Check this result by multiplying (x 2 4x + 6) by 2. When you do, you will obtain the original polynomial. 14
15 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 2x 2 + 8x 12, you could simply write 2x 2 + 8x 12 = 2(x 2 4x + 6). 15
16 Factoring by Grouping 16
17 Factoring by Grouping There are occasions when the common factor of an expression is not simply a monomial. For instance, the expression x 2 (x 2) + 3(x 2) has the common binomial factor (x 2). Factoring out this common factor produces x 2 (x 2) + 3(x 2) = (x 2)(x 2 + 3). This type of factoring is part of a more general procedure called factoring by grouping. 17
18 Example 9 Common Binomial Factors Factor each expression. a. 5x 2 (7x 1) 3(7x 1) b. 2x(3x 4) + (3x 4) c. 3y 2 (y 3) + 4(3 y) Solution: a. Each of the terms of this expression has a binomial factor of (7x 1). 5x 2 (7x 1) 3(7x 1) = (7x 1)(5x 2 3) 18
19 Example 9 Common Binomial Factors b. Each of the terms of this expression has a binomial factor of (3x 4). cont d 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. 3y 2 (y 3) + 4(3 y) = 3y 2 (y 3) 4(y 3) = (y 3)(3y 2 4) Write 4(3 y) as 4(y 3). Common factor is (y 3). 19
20 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 x 3 + 2x 2 + 3x + 6 and try to factor it. Note first that there is no common monomial factor to take out of all four terms. 20
21 Factoring by Grouping But suppose you group the first two terms together and the last two terms together. x 3 + 2x 2 + 3x + 6 = (x 3 + 2x 2 ) + (3x + 6) = x 2 (x + 2) + 3(x + 2) = (x + 2)(x 2 + 3) Group terms. Factor out common monomial factor in each group. Factored form 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 x 3 with the fourth term 6. 21
22 Example 10 Factoring by Grouping Factor x 3 + 2x 2 + x + 2. Solution: x 3 + 2x 2 + x + 2 = (x 3 + 2x 2 ) + (x + 2) = x 2 (x + 2) + (x + 2) = (x + 2)(x 2 + 1) Group terms. Factor out common monomial factor in each group. Factored form 22
23 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. x 3 + 2x 2 + x + 2 = (x 3 + x) + (2x 2 + 2) = x(x 2 + 1) + 2(x 2 + 1) = (x 2 + 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. 23
24 Example 12 Geometry: Area of a Rectangle The area of a rectangle of width (2x 1) feet is (2x 3 + 2x x 2 ) square feet, as shown below. Factor this expression to determine the length of the rectangle. 24
25 Example 12 Geometry: Area of a Rectangle Solution Verbal Model: cont d Labels: Area = 2x 3 + 2x x 2 (square feet) Width = 2x 1 Equation: 2x 3 + 4x x 2 2 = (2x 3 + 4x) + ( x 2 2) Group terms. (feet) = 2x(x 2 + 2) + (x 2 + 2) = (x 2 + 2)(2x 1) Factor out common monomial factor in each group. Factored form The length of the rectangle is (x 2 + 2) feet. 25
26 Homework: Page 272 # down Page 273 # 45, 47, 49 Page 274 # down
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