Problem 1 Oh Snap... Look at the Denominator on that Rational

Transcription

1 Problem Oh Snap... Look at the Denominator on that Rational Previously, you learned that dividing polynomials was just like dividing integers. Well, performing operations on rational epressions involving variables is just like performing operations on rational numbers. Rational Numbers Rational Epressions Involving Variables in the Numerator Eample Eample 2 Eample () 2() 2(4) 2 (5) 5(4) 4(5) 2y 2 5 () 2(4)y 2 (5) () 5(4) 4(5) y y () 5() 2(5) 2 2 (5) 2y y 5 () 2(5)y y)() (2 5 5() (5) 5() y y y (6 9y) 9 y y 5 5 y 2 2. Analyze the eamples. a. Eplain the process used to add and subtract each epression. b. In Eample 2, why is 5 () 2 2() and why is 5 () 2 2()? (2 y)() c. In Eample, eplain how y. 5() 72 Chapter Solving Rational Equations

2 2. Analyze Noelle s work. Noelle _ 2 To determine a common denominator, I multiply all the denominators together:? 8? (6) (6) 2(6) 2 (24) 8(6) 2(24) Eplain how Noelle could have added the rational epressions more efficiently.. Calculate each sum and difference. a y 8 b. 2 2y 2 2 z 4 c Is the set of rational epressions closed under addition and subtraction? Eplain your reasoning.. Adding and Subtracting Rational Epressions 7

3 5. Notice that all the variables in the right column of the table are in the numerator. If there were variables in the denominator, do you think the process of adding and subtracting the epressions would change? Eplain your reasoning Problem 2 Umm, I Think There Are Some Variables in Your Denominator... When rational epressions contain variables in the denominator, the process remains the same you still need to determine the least common denominator (LCD) before adding and subtracting. It will save time and effort if you determine the LCD and use it to add and subtract rational epressions.. Consider Method A compared to Method B in both columns of the table. All your factoring skills will come in handy. Rational Numbers Rational Epressions Involving Variables in the Denominator Method A Method B 5 ) 2 (2 () ( 2 ) 2 () ( ) 5 ( 2 ) () () 2 5 ) 2 (2 () ( 2 ) 2 () ( ) ( 2 ) for fi () 2 () for fi Chapter Solving Rational Equations

4 How are these restriction(s) shown in the graph of the function? a. Eplain the difference in the methods. Which method do you prefer? b. Eplain why the statement has the restriction fi 0.? 2. Ruth and Samir determine the LCD for the epression: Ruth 2 2 ( 2 2 )( ) Samir 2 2 ( 2 )( ) ( 2 )( ) 2 2 a. Who is correct? Eplain your reasoning. b. Describe any restriction(s) for the value of.. Adding and Subtracting Rational Epressions 7

5 . Calculate the least common denominator for each pair of rational epressions. a., Make sure to list the restrictions for the variable. b , c. 2, 7 2 4, d , Notice that even though there are binomials in the denominator, adding two rational epressions is similar to adding two rational numbers. Rational Epressions Involving Binomials in the Denominator ( )( 2 ) 2 5 ( )( 2 ) 2 ( 2 ) ( )( 2 ) 5 ( )( 2 ) 2 2 ( )( 2 ) 5 2 ( )( 2 ) ( )( 2 ) 76 Chapter Solving Rational Equations

6 4. Marissa and Salvatore add 2 2. Marissa 2 2 _ 5 2)() (2 ) ( ( )() ( ) ( )() ( ) ( ) 5 22 ( ) (2 )( ) 5 ( ) 5 2 Salvatore 2 2 _ 5 ) 2( ( ) _ 5 2 _ 5 2() () _ 5 2 Eplain the difference in the methods used.? 5. Randy says the only restriction on the variable in Marissa and Salvatore s problem is fi 0. Cynthia says fi 0, 2. Who is correct? Eplain your reasoning.. Adding and Subtracting Rational Epressions 77

7 6. Calculate each sum or difference. Make sure to list the restrictions for the variable, and simplify when possible. a b c d Chapter Solving Rational Equations