7. Rational Epressions - Comple Fractions Objective: Simplify comple fractions by multiplying each term by the least common denominator. Comple fractions have fractions in either the numerator, or denominator, or usually both. These fractions can be simplified in one of two ways. This will be illustrated first with integers, then we will consider how the process can be epanded to include epressions with variables. The first method uses order of operations to simplify the numerator and denominator first, then divide the two resulting fractions by multiplying by the reciprocal. Eample. 6 + 8 6 + 6 Get common denominator in top and bottom fractions Add and subtract fractions, reducing solutions ( ( )( ) )( ) To divide fractions we multiply by the reciprocal Reduce Multiply 6 The process above works just fine to simplify, but between getting common denominators, taking reciprocals, and reducing, it can be a very involved process. Generally we prefer a different method, to multiply the numerator and denominator of the large fraction (in effect each term in the comple fraction) by the least common denominator (LCD). This will allow us to reduce and clear the small fractions. We will simplify the same problem using this second method. Eample. 6 + LCD is, multiply each term
() () () + () 6 () () ()+(6) 8 0 + 6 6 Reduce each fraction Multiply Add and subtract Clearly the second method is a much cleaner and faster method to arrive at our solution. It is the method we will use when simplifying with variables as well. We will first find the LCD of the small fractions, and multiply each term by this LCD so we can clear the small fractions and simplify. Eample. LCD = Identify LCD (use highest eponent) Multiply each term by LCD ( ) ( ) ( ) ( ) ( ) ( ) ( + )( ) ( ) + Reduce fractions (subtract eponents) Multiply Factor Divide out( ) factor The process is the same if the LCD is a binomial, we will need to distribute + + + Multiply each term by LCD, ( +)
( + ) + ( + )+ ( + ) ( + ) + ( + ) ( +) + 8 + 0 + Reduce fractions Distribute Combine like terms + The more fractions we have in our problem, the more we repeat the same process. Eample. + ab ab ab a b +ab ab LCD =a b Idenfity LCD (highest eponents) Multiply each term by LCD (a b ) (a b ) + (a b ) ab ab ab (a b ) a b + ab(a b ) (a b ) ab ab a + ab b +a b ab Reduce each fraction(subtract eponents) World View Note: Sophie Germain is one of the most famous women in mathematics, many primes, which are important to finding an LCD, carry her name. Germain primes are prime numbers where one more than double the prime number is also prime, for eample is prime and so is + = 7 prime. The largest known Germain prime (at the time of printing) is 807 60 which has 799 digits! Some problems may require us to FOIL as we simplify. To avoid sign errors, if there is a binomial in the numerator, we will first distribute the negative through the numerator. Eample. + + + + + + + + + + Distribute the subtraction to numerator Identify LCD
LCD =( + )( ) Multiply each term by LCD ( )( + )( ) ( )(+)( ) + + ( )(+)( ) ( + )( + )( ) + + ( )( )+( )( + ) ( )( )+( +)( + ) 6 +9 6 9 6 + 9+ +6 9 + 8 ( + 9) 6 9 Reduce fractions FOIL Combine like terms Factor out in denominator Divide out common factor If there are negative eponents in an epression we will have to first convert these negative eponents into fractions. Remember, the eponent is only on the factor it is attached to, not the whole term. Eample 6. m + m m +m Make each negative eponent into a fraction m + m m + m Multiply each term by LCD,m (m ) + (m ) m m m(m )+ (m ) m +m m + Reduce the fractions Once we convert each negative eponent into a fraction, the problem solves eactly like the other comple fraction problems. Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/)
7. Practice - Comple Fractions Solve. ) + ) y + y ) a a a ) a a +a ) a a a + a 6) b + b 7) + 0 + 8) + + 9) a + 6 a 0) b 0 b +6 ) + + + ) a a a 6 a ) 9 ) 9 ) a b a b a+b 6ab 6) 6 + 7) 0 + + 8 8) + 9) ) + + + 6 0) +0 8 + 0 ) 8 + + 7+ 6 +
) + 9 + + + ) a a + a a ) b b + + b b + 6) y y y + y 7) b ab a b + 7 ab + a 8) + + + + + 9) y y y + y y + y y + y 0) + + ( + ) + ( ) Simplify each of the following fractional epressions. ) y + y ) y y y ) y +y y ) + ) 6 +9 9 6) + y y + y Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/) 6
7. ) ) y y ) a a+ ) a a ) a a+ 6) b + b b 8b 7) 8) 9) 0) ) + + ) a a + a a ) Answers - Comple Fractions ) + ) b(a b) a 6) + 7) + 9 8) 9) 0) +8 + ) + ) 7 + ) + ) a a ) b b + 6) + y y ( )( + ) + 7) a b a+b 8) + 9) y 0) ) y y ) y + y y ) + y y ) + ) ( ) ( + )( ) 6) + y y Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/) 7