The degree of control a kayaker exerts over the kayak

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College Algebra
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Chapter 1 / Exercise 41
College Algebra
Larson
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The degree of control a kayaker exerts over the kayak depends largely on the body con- tact with it. A kayaker wears the kayak. So the choice of a kayak should hinge first on the right body fit and comfort and second on the skill level or intended paddling style. So design- ing, building, and even fitting a kayak is a blend of art and science. Math at Work Kayak Design In This Section U 1 V Factoring by Grouping U 2 V Factoring a Difference of Two Squares U 3 V Factoring a Perfect Square Trinomial U 4 V Factoring Completely 5.2 Special Products and Grouping In Section 5.1 you learned how to factor out the greatest common factor from all of the terms of a polynomial. In this section you will learn to factor a four-term polynomial by factoring out a common factor from the first two terms and then a common factor from the last two terms. U 1 V Factoring by Grouping The product of two binomials may have four terms. For example, ( x a )( x 3) ( x a ) x ( x a )3 x 2 ax 3 x 3 a . To factor x 2 ax 3 x 3 a , we simply reverse the steps we used to find the product. Factor out the common factor x from the first two terms and the common factor 3 from the last two terms: x 2 ax 3 x 3 a x ( x a ) 3( x a ) Factor out x and 3. ( x 3)( x a ) Factor out x a . It does not matter whether you take out the common factor to the right or left. So ( x a )( x 3) is also correct and we could have factored as follows: x 2 ax 3 x 3 a ( x a ) x ( x a )3 ( x a )( x 3) 330 Chapter 5 Factoring 5-10
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College Algebra
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Chapter 1 / Exercise 41
College Algebra
Larson
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5-11 5.2 Special Products and Grouping 331 This method of factoring is called factoring by grouping. Strategy for Factoring a Four-Term Polynomial by Grouping 1. Factor out the GCF from the first group of two terms. 2. Factor out the GCF from the last group of two terms. 3. Factor out the common binomial. Factoring by Grouping Use grouping to factor each polynomial. a) xy 2 y 5 x 10 b) x 2 wx x w Solution a) The first two terms have a common factor of y , and the last two terms have a common factor of 5: xy 2 y 5 x 10 y ( x 2 ) 5( x 2 ) Factor out y and 5. ( y 5)( x 2 ) Factor out x 2. Check by using FOIL. b) The first two terms have a common factor of x , and the last two have a common factor of 1: x 2 wx x w x ( x w ) 1( x w ) Factor out x and 1. ( x 1)( x w ) Factor out x w . Check by using FOIL. Now do Exercises 1–10 E X A M P L E 1 E X A M P L E 2 Factoring by Grouping with Rearranging Use grouping to factor each polynomial. a) mn 4 m m 2 4 n b) ax b bx a Solution a) We can factor out m from the first two terms to get m ( n 4), but we can’t get another factor of n 4 from the last two terms. By rearranging the terms we can factor by grouping: mn 4 m m 2 4 n m 2 mn 4 m 4 n Rearrange terms. m ( m n ) 4( m n ) Factor out m and 4. ( m 4)( m n ) Factor out m n . For some four-term polynomials it is necessary to rearrange the terms before fac- toring out the common factors.
332 Chapter 5 Factoring 5-12 Note that there are several rearrangements that will allow us to factor the polynomi- als in Example 2. For example, m 2 4 m mn 4 n would also work for Example 2(a).

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