Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators over that denominator: 9 6 9 6 8 8 8 8 If they have different denominators, you have to create a common denominator, then combine the numerators: () ( 4 5 4 5 9 5 5() ( 0 0 0 0 And, once you complete the operation, always check to see if you can simplify the result: i 6 6 6 i Adding and subtracting rational epressions works eactly the same way you just have some variables involved and, perhaps, more comple factors. If the denominators look eactly the same, no matter how comple they are, then you can combine the numerators over that denominator. If the denominators are different, then you have to find a common denominator and modify the epressions before doing any operations with the numerators. Same denominators: Combine numerators and simplify, if possible 5 5 8 5 ( ( ) 5 5 4 4 4 4 4 ) 4 ( ) ( ) Department of Mathematics, Sinclair Community College, Dayton, OH
Same denominators ecept for sign: Remember in 6. how we changed opposite factors into the same factor by factoring out a negative? If you have denominators that are the same ecept for opposite signs, you can make them the same denominator by doing the same thing. The that is factored out can be moved around, like to the numerator. Remember: a a a b b b sign moved ) 5 5 5 5 y 4 4 y y 4 ( 4 y) y 4 y 4 y 4 y 4 factor out a move the negative switch the order sign to the numerator Different denominators: Just like with fractions, if the denominators of our rational epressions are different, we have to make them the same in order to add or subtract the epressions and we want a least common denominator (LCD) (that is, with the fewest number of factors). A common denominator contains within it all the factors of each denominator. Rule. If you re looking at simple products of variables, like 5 y eponent of each variable in the common denominator: and y 4, then use the highest 5 y y 4 5 y 4 LCD Rule. If you have numerical coefficients as well as variables, like 6, y, and 4, factor out any number common to all the coefficients first, then see what s left. Use this only once! Then, see if you have a factor common to the remaining terms use this factor only once (continue as needed). Then, include any other factors left over from the different denominators. Remember, the common denominator must include all factors of each denominator within it. 6 y 4 6 6 y 6 common to two additional factor (to all three) 6 y 4 y LCD highest powers of variables Department of Mathematics, Sinclair Community College, Dayton, OH
Rule. If you re dealing with polynomials, like 4, you must first factor the polynomials of each denominator (if possible). Then use those factors to build a common denominator. 4 8 ( )( 6) ( ( ) ( )( 6)( LCD After you find the LCD, you ll have to multiply the numerator and denominator of each rational epression by whatever factors are missing compared to the common denominator. For eample, to convert and bottom by : y to an epression with an LCD of 4 y ( y ), multiply top y 4 y To convert to an epression with an LCD of ( 6)( ( ), multiply top and ( 6)( ) ( ( )( bottom by the missing factor ( :. ( 6)( ) ( ( 6)( )( Do not be tempted to simplify these epressions again! You re creating them so you can complete the other operations of addition and subtraction with the numerators. Eample. Perform the indicated operation and simplify: y y Step. Do the fractions have the same denominator? Yes! y y Step. Add the numerators over the common denominator. Step. Can the resulting epression be simplified? No. Done! y y 4y Department of Mathematics, Sinclair Community College, Dayton, OH
Eample. Perform the indicated operation and simplify: 5 Step. Do the fractions have the same denominator? Yes! Step. Add the numerators over the common denominator. 5 Don t forget to distribute that minus sign! 5 Step. Can the resulting epression be simplified? Well, 5 ( ( ) ( ) let s try factoring the numerator and see what we get. ( ( ) ( ) 5 ( Done! Eample. Perform the indicated operation and simplify: 4 9 6 Step. Do the fractions have the same denominator? Not quite! Step. Find the LCD. First factor the denominators, 9 6 then include all the factors of each denominator in a common denominator highest power of the variable Step. Multiply top and bottom of each rational epression by whatever factor is missing in the denominator compared to the LCD. 8 LCD 4 i i 9 6 ( ( 8 8 Step 4. Add the numerators over the common denominator. Combine like terms. ( ( 8 6 8 6 8 8 0 8 Department of Mathematics, Sinclair Community College, Dayton, OH 4
Step 5. Can the resulting epression be simplified? (4 i9 Try factoring. (4 4 5 i9 9 Done. Eample 4. Perform the indicated operation and simplify: Step. Do the fractions have the same denominator? 4 No. Step. Find the LCD. First factor the denominators, then make sure all the factors of each denominator are ( ( ( included within a common denominator. ( ( LCD Step. Multiply top and bottom of each rational epression by whatever factor is missing in the denominator compared to the LCD. 4 ( i i ( ( ( ( 4( ( ( ( ( Step 4. Add the numerators over the common denominator. Step 5. Can the resulting epression be simplified? Try factoring. 4( ( ( 4 4 ( ( ( 4 ( ( ( )( ) ( ( ( ) ( ( ( ) ( ( Done. Department of Mathematics, Sinclair Community College, Dayton, OH 5
Problems Perform the indicated operation and simplify, if possible. 5 6) 4 5 5 8 6 7) a b ab 0 9 4 ) ) a 6 a 6 8) 6 4 4 9) 6z 9 y z y z z y 5 5 0) m m m Answers 5 6) 7 0 ( ( ) 8b 6a a b 7) ( 7)( )( ) a 8) ( )( 9) 6 y z 0 ( ( 0) m m m (m Department of Mathematics, Sinclair Community College, Dayton, OH 6