Section 2.4: Add and Subtract Rational Expressions
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1 CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall that when adding fractions with a common denominator, we add the numerators and keep the denominator. This same process is used with rational expressions. Remember to reduce your sum or difference, if possible, to obtain your final answer. ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH THE SAME DENOMINATOR Example 1. Add the rational expressions, and simplify if possible. x x x 8 x 8 x x 8 x x x 8 ( x ) ( x)( x) Same denominator; add numerators; combine like terms Factor numerator and denominator Divide out common factor of ( x ) to reduce the fraction to its lowest terms x Our Answer Subtraction with common denominators follows the same pattern. However, subtraction can cause problems if we are not careful to avoid sign errors. Consequently, we will first distribute the subtraction sign to every term in the numerator of the fraction that follows the subtraction sign. Then we can treat it like an addition problem. Example. Subtract the rational expressions, and simplify if possible. 6x1 1x6 x 6 x 6 6x 1 1x 6 x6 x6 9x 6 x 6 Add the opposite of the second fraction (distribute the subtraction to each term in the second fraction) Add the numerators; combine like terms Factor numerator and denominator Page 79
2 CHAPTER Section.: Add and Subtract Rational Expressions (x ) ( x ) (x ) x Divide out common factor of Our Answer ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH DIFFERENT DENOMINATORS When the denominators of the rational expressions are not like, we find their least common denominator (LCD). Then we build up each fraction to an equivalent one with that LCD as the denominator. The following example shows this process with fractions. Example. Add the fractions, and simplify if possible. 1 The LCD is 1. Build up, multiply 6 by and by 6 to get the common denominator; 1 6 Multiply first fraction by and second by Add the numerators 1 1 Our Answer The same process is used with rational expressions containing variables. Example. Add the rational expressions, and simplify if possible. 7a b The LCD is 6a b. Build up each expression: a b 6ab ( a b)( b ) 6a b and (6 ab )( a) 6a b 7a b b a a b b 6ab a b Multiply first fraction by b and second by a a Page 80
3 CHAPTER Section.: Add and Subtract Rational Expressions 1ab ab 6a b 6a b Add the numerators, there are no like terms to combine 1ab ab Factor the numerator 6a b ab(7 b ) Reduce, dividing out common factors of, a, and b 6a b 7b Our Answer ab The same process is used for subtraction; we will simply include the first step of changing the subtraction to addition of the opposite value. Example. Subtract the rational expressions, and simplify if possible. 7b a a Change subtraction to addition of the opposite 7b a a The LCD is ( a)( a) 0a 0a. Build up each expression: and ( a )() 0a a 7b a a a Multiply first fraction by a a and second by 16a b 0a 0a Add numerators; there are no like terms to combine 16a b Factor the numerator, if possible. In this case, the 0a numerator is prime. This is our answer. If our denominators have more than one term, then we will need to factor first to find the LCD. Next, we build up each fraction using the factors that are missing from each denominator. Example 6. Add the rational expressions, and simplify if possible. 6 a 8a 8 Factor denominators to find LCD Page 81
4 CHAPTER Section.: Add and Subtract Rational Expressions 6 a (a 1) 8 The LCD is 8(a 1). Build up each expression: (a1) 8(a 1) and (8)(a 1) 8(a 1) 6 a a1 (a1) 8 a1 1 6a a 8(a1) 8(a1) 6a a 1 8(a 1) (a a ) 8(a 1) (a a ) 8(a 1) Multiply first fraction by Add numerators Factor the numerator Cannot be simplified Our Answer, second by a 1 a 1 Example 7. Add the rational expressions, and simplify if possible. y Factor denominators to find LCD y 0 6y y y ( y ) (6y 1)( y ) y ( y ) (6y 1)( y ) 6y1 y ( y ) 6y 1 (6y 1)( y ) The LCD is ( y) (6y 1). Build up each expression 6y 1 Multiply first fraction by and 6y 1 second by Multiply the numerators and denominators 18y y10 ( y )(6y 1) (6y 1)( y ) Add numerators, combine like terms y 7 ( y)(6y1) Factor the numerator, if possible. In this case, the numerator is prime. This is our answer. Page 8
5 CHAPTER Section.: Add and Subtract Rational Expressions Whenever you encounter a subtraction problem, remember to rewrite the problem using addition of the opposite. Example 8. Subtract the rational expressions, and simplify if possible. x1 x1 x x 7x 1 Add the opposite: Distribute the subtraction to each term in the numerator of the second fraction x1 x1 x x 7x 1 Factor denominators to find LCD x1 x1 x ( x )( x ) The LCD is ( x) ( x ). Build up each expression x 1 x x 1 x x ( x )( x ) Only the first fraction needs to be multiplied by ( x ) x x x1 ( x )( x ) ( x )( x ) Add the numerators, combine like terms x x ( x )( x ) Factor the numerator ( x)( x1) ( x)( x) Divide out the common factor of ( x ) x 1 x Our Answer Page 8
6 CHAPTER Section.: Add and Subtract Rational Expressions Practice Exercises Section.: Add and Subtract Rational Expressions Add or subtract, expressing the result in its lowest terms. 1) a a a1 a1 1) a 9a ) x 6x8 x x 1) x1 x x x ) ) t t t 7 t1 t1 a a a a 6 a a 6 c d c d 1) c d cd x y x y 1) x y xy ) x 6) x x 7) 6r 8r x x x 9 6x x 6x 7 xy x y 8) 8 9t 6t 9) 10) 11) x x 8 1 a a 16) x1 x1 17) z z z1 z1 18) x x 8 19) x x x x 0) x x t 1) t t1 ) x ( x) The Practice Exercises are continued on the next page. Page 8
7 CHAPTER Section.: Add and Subtract Rational Expressions Practice Exercises: Section. (continued) ) x x x x 1 x ) x x x x ) ) a 9a a0 6a0 t y y t y t x1 x 6 ) x x x 7x10 x x 6) x6 x 6) x x x x 7) a a a 9 a x 7) x x 6 x x x 8) x 1 x x x 7 9) x 1x 6 x 1x x 0) x 9 x x 6 1) x 18 x x 6 x 9 x ) x x x x 6 x ) x 1 x x y 8) y 1 y y 1 z z 9) 1 z z 1 z 1 0) r 1 1 r s r s r s x x1 1) x x x x 6 x x ) x x x x x 7 x ) x x x 6x x8 x ) x 6x8 x x Page 8
8 CHAPTER Section.: Add and Subtract Rational Expressions ANSWERS to Practice Exercises Section.: Add and Subtract Rational Expressions 1) 6 a 1) a 7a 9a ) x ) t 7 ) ) a a 6 x 6 x x 6) x 1) 1) 1) 16) 7x 1 x c y cd d c d xy 6x x y x ( x1)( x1) 7) r 17) z z ( z1)( z1) 7x y x y 8) 1t 16 9) 18t 10) x 9 11) a 8 18) 19) 0) 1) 11x 1 xx ( ) 1 x ( x)( x) x x ( x )( x ) t ( t ) x 10 ) ( x ) The Answers to Practice Exercises are continued on the next page. Page 86
9 CHAPTER Section.: Add and Subtract Rational Expressions ANSWERS to Practice Exercises: Section. (continued) 6 0x ) 1 xx ( 1) ) x 7 ( x)( x) ) 9a ( a ) ) x 8 ( x7)( x) ) t ty y ( yt)( y t) 6) x 7x ( x)( x) 6) x 10x x( x ) 7) a ( a)( a) 7) x ( x)( x1) 8) yy ( 1) 8) x ( x1)( x) 9) z z 1 9) x 8 ( x8)( x6) 0) r s 0) x ( x)( x) 1) ( x 1) ( x1)( x) 1) x 1 ( x)( x) ) x ( x)( x ) ) x 1 ( x1)( x) ) ( x 9) ( x)( x) ) x ( x)( x1) ) x 10 ( x)( x1) Page 87
10 CHAPTER Section.: Add and Subtract Rational Expressions Page 88
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