6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as Because 2 3


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1 0 (618) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) Onehalf of 1 b) Onethird of c) Onehalf of d) Onehalf of 1 a) b) c) d) 8 7. Eploration. Let R 6 0 and H a) Find R when and. Find H when and. b) How are these values of R and H related and why a) R, R 1 1, H, H 1 1 In this section Building Up the Denominator Finding the Least Common Denominator 6. FINDING THE LEAST COMMON DENOMINATOR Every rational epression can be written in infinitely many equivalent forms. Because we can add or subtract only fractions with identical denominators, we must be able to change the denominator of a fraction. You have already learned how to change the denominator of a fraction by reducing. In this section you will learn the opposite of reducing, which is called building up the denominator. Building Up the Denominator To convert the fraction factor 1 as 1 7. Because into an equivalent fraction with a denominator of 1, we already has a in the denominator, multiply the numerator and denominator of by the missing factor 7 to get a denominator of 1: For rational epressions the process is the same. To convert the rational epression into an equivalent rational epression with a denominator of 1, first factor 1: 1 ( )( ) From the factorization we can see that the denominator needs only a factor of to have the required denominator. So multiply the numerator and denominator by the missing factor : ( ) ( )( ) 0 1 E X A M P L E 1 Building up the denominator Build each rational epression into an equivalent rational epression with the indicated denominator. a) b) w c) 1 w y 1y 8
2 6. Finding the Least Common Denominator (619) 1 a) Because, we get a denominator of 1 by multiplying the numerator and 1 denominator by 1: b) Multiply the numerator and denominator by : w w w c) To build the denominator y up to 1y 8, multiply by y : y y 8y y y 1 y 8 In the net eample we must factor the original denominator before building up the denominator. E X A M P L E helpful hint Notice that reducing and building up are eactly the opposite of each other. In reducing you remove a factor that is common to the numerator and denominator, and in building up you put a common factor into the numerator and denominator. Building up the denominator Build each rational epression into an equivalent rational epression with the indicated denominator. 7 a) b) y 6y a) Because y ( y), we factor 6 out of 6y 6. This will give a factor of y in each denominator: y ( y) 6y 6 6( y) ( y) To get the required denominator, we multiply the numerator and denominator by only: 7 y 1 6y 6 b) Because 8 1 ( )( 6), we multiply the numerator and denominator by 6, the missing factor: 7() ( y)() ( )( 6) ( )( 6) CAUTION When building up a denominator, both the numerator and the denominator must be multiplied by the appropriate epression, because that is how we build up fractions.
3 (60) Chapter 6 Rational Epressions study tip Studying in a quiet place is better than studying in a noisy place. There are very few people who can listen to music or a conversation and study at the same time. Finding the Least Common Denominator We can use the idea of building up the denominator to convert two fractions with different denominators into fractions with identical denominators. For eample, 6 and can both be converted into fractions with a denominator of 1, since 1 6 and 1 : The smallest number that is a multiple of all of the denominators is called the least common denominator (LCD). The LCD for the denominators 6 and is 1. To find the LCD in a systematic way, we look at a complete factorization of each denominator. Consider the denominators and 0: 0 Any multiple of must have three s in its factorization, and any multiple of 0 must have one as a factor. So a number with three s in its factorization will have enough to be a multiple of both and 0. The LCD must also have one and one in its factorization. We use each factor the maimum number of times it appears in either factorization. So the LCD is : 10 If we omitted any one of the factors in, we would not have a multiple of both and 0. That is what makes 10 the least common denominator. To find the LCD for two polynomials, we use the same strategy. 1 0 Strategy for Finding the LCD for Polynomials 1. Factor each denominator completely. Use eponent notation for repeated factors.. Write the product of all of the different factors that appear in the denominators.. On each factor, use the highest power that appears on that factor in any of the denominators. E X A M P L E Finding the LCD If the given epressions were used as denominators of rational epressions, then what would be the LCD for each group of denominators a) 0, 0 b) yz, y z, yz c) a a 6, a a
4 6. Finding the Least Common Denominator (61) a) First factor each number completely: 0 0 The highest power of is, and the highest power of is. So the LCD of 0 and 0 is, or 100. b) The epressions yz, y z, and yz are already factored. For the LCD, use the highest power of each variable. So the LCD is y z. c) First factor each polynomial. a a 6 (a )(a ) a a (a ) The highest power of (a ) is 1, and the highest power of (a ) is. So the LCD is (a )(a ). E X A M P L E helpful hint What is the difference between LCD, GCF, CBS, and NBC The LCD for the denominators and 6 is 1. The least common denominator is greater than or equal to both numbers. The GCF for and 6 is. The greatest common factor is less than or equal to both numbers. CBS and NBC are TV networks. When adding or subtracting rational epressions, we must convert the epressions into epressions with identical denominators. To keep the computations as simple as possible, we use the least common denominator. Find the LCD for the rational epressions, and convert each epression into an equivalent rational epression with the LCD as the denominator. a) 9 y, b) 1z 6, 1, 8 y y a) Factor each denominator completely: 9y y 1z z The LCD is yz. Now convert each epression into an epression with this denominator. We must multiply the numerator and denominator of the first rational epression by z and the second by y: 9y z 0z 9y z yz y 6y 1z 1 z y yz Same denominator b) Factor each denominator completely: 6 8 y y y y The LCD is y or y. Now convert each epression into an epression with this denominator: 6 y 0y 6 y y 1 1 y y 8 y 8 y y y y 6 18 y 6 y
5 (6) Chapter 6 Rational Epressions E X A M P L E Find the LCD for the rational epressions and 6 and convert each into an equivalent rational epression with that denominator. First factor the denominators: ( )( ) 6 ( )( ) The LCD is ( )( )( ). Now we multiply the numerator and denominator of the first rational epression by ( ) and those of the second rational epression by ( ). Because each denominator already has one factor of ( ), there is no reason to multiply by ( ). We multiply each denominator by the factors in the LCD that are missing from that denominator: ( ) ( )( )( ) 6 ( ) ( )( )( ) Note that in Eample we multiplied the epressions in the numerators but left the denominators in factored form. The numerators are simplified because it is the numerators that must be added when we add rational epressions in the net section. Because we can add rational epressions with identical denominators, there is no need to multiply the denominators. 1 ( )( )( ) 6 ( )( )( ) Same denominator WARMUPS True or false Eplain your answer. 1. To convert into an equivalent fraction with a denominator of 18, we would multiply only the denominator of by 6. False. Factoring has nothing to do with finding the least common denominator. False. a b a b for any nonzero values of a and b. 10a b True. The LCD for the denominators and is. True. The LCD for the fractions 1 6 and 1 is False 6. The LCD for the denominators 6a b and ab is ab. False 7. The LCD for the denominators a 1 and a 1isa 1. False 7 8. for any real number. False The LCD for the rational epressions and is. 10. for any real number. True True
6 6. Finding the Least Common Denominator (6) 6. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is building up the denominator We can build up a denominator by multiplying the numerator and denominator of a fraction by the same nonzero number.. How do we build up the denominator of a rational epression To build up the denominator of a rational epression, we can multiply the numerator and denominator by the same polynomial.. What is the least common denominator for fractions For fractions, the LCD is the smallest number that is a multiple of all of the denominators.. How do you find the LCD for two polynomial denominators For polynomial denominators, the LCD consists of every factor that appears, raised to the highest power that appears on the factor. Build each rational epression into an equivalent rational epression with the indicated denominator. See Eample y y y 9. b 1t bt ay 7z bt ayz ayz 11. 9z 6z aw 8a wz 8aw z 1. 7yt y t 18 yt 18yt a 7b 1. a 1a 1 c 6 c 1bc 8 8 1a 6c 1. 8y y y z y 10 y 1 0 y 8 z z z Build each rational epression into an equivalent rational epression with the indicated denominator. See Eample m n n m n m 8a ab 19. b b 0b 0b 0b 0b a a a. a a 9 a y y 6 y 0 y y y 0 y y 0 6. z z 6 11z 0 z z z 1 z z 1 If the given epressions were used as denominators of rational epressions, then what would be the LCD for each group of denominators See Eample. 7. 1, , , 18, , 0, a,1a 0a. 18,0y 180 y. a b,ab 6, a b 1a b 6. m nw, 6mn w 8,9m 6 nw 6m 6 n w 8. 16, 8 16 ( )( ) 6. 9, 6 9 ( )( ) 7.,, ( )( ) 8. y, y, y y(y )(y ) 9., 16, ( )( ) 0. y, y y, y y(y ) Find the LCD for the given rational epressions, and convert each rational epression into an equivalent rational epression with the LCD as the denominator. See Eample , 8 9,., 9, , 6 b 9b 0a, 8a ab ab b 6 8b 0.,, 7 a 10 ab ab ab., 1 9 6, 6 9 c 0ab 6.,, 8a b 9 6a c a 9b 9 c a 9 b c 6 1 y z 6y 7.,,, 9y z 1 y z, 6 y 6 y z 6 y z 6 y z b b 18a b a 8.,,,, 1 a 6 b a 1b 8 a 6 b 8 6 1a a b 8 a 6 b In Eercises 9 60, find the LCD for the given rational epressions, and convert each rational epression into an equivalent rational epression with the LCD as the denominator. See Eample. 1 9.,, ( ) ( ) ( )( )
7 6 (6) Chapter 6 Rational Epressions a a a a a 1a 0.,, a a (a ) (a ) (a )(a ) 1.,, a 6 6 a a 6 a 6 8.,, y y y y 1.,, ( ) ( ) ( ) ( ).,, 1 1 ( 1) ( 1) ( 1)( 1) w. w w, w 1 w w w w w 6w, (w )(w )(w 1) (w )(w )(w 1) z 1 z 1 6., z 6z 8 z z 6 z z z z, (z )(z )(z ) (z )(z )(z ) 7.,, ,, 6( ) ( ) 6( )( ) 6( )( ) b 8.,, b 9 b b b b b 6b, b(b ) (b ) b(b )(, b ) 10b 1 b(b ) (b ) 9.,, q q q 9q q q 1 q 8 q 9,, (q 1)(q )(q ) (q 1)(q )(q ) 8q (q 1)(q )(q ) p 60.,, p 7p 1 p 11p 1 p p 0 p 1 p p,, (p )(p )(p ) (p )( p )(p ) p 6 (p )(p )(p ) GETTING MORE INVOLVED 61. Discussion. Why do we learn how to convert two rational epressions into equivalent rational epressions with the same denominator Identical denominators are needed for addition and subtraction. 6. Discussion. Which epression is the LCD for 1 7 and ( ) ( ) a) ( ) b) 6( ) c) 6 ( ) d) ( ) c 6. ADDITION AND SUBTRACTION In this section In Section 6. you learned how to find the LCD and build up the denominators of rational epressions. In this section we will use that knowledge to add and subtract rational epressions with different denominators. Addition and Subtraction of Rational Numbers Addition and Subtraction of Rational Epressions Applications Addition and Subtraction of Rational Numbers We can add or subtract rational numbers (or fractions) only with identical denominators according to the following definition. Addition and Subtraction of Rational Numbers If b 0, then a c a c a c a c and. b b b b b b
FINDING THE LEAST COMMON DENOMINATOR
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