Complex fraction: - a fraction which has rational expressions in the numerator and/or denominator

Transcription

1 Comple fraction: - a fraction which has rational epressions in the numerator and/or denominator o y 2 + y 2 y 2 2 Steps for Simplifying Comple Fractions. simplify the numerator and/or the denominator by adding and/or subtracting the rational epressions 2. use the procedure for dividing fractions to change division to multiplication 3. factor the numerator and denominator completely 4. simplify the fraction completely by canceling common factors y 2+ y 2 y y 3 2 y 2 2 y 2 2 y 2 3 +y 3 2 y 2 2 y 2 2 y 2 3 +y 3 2 y 2 2 y 2 2 y 2 (+y)( 2 y+y 2 )( 2 y 2 ) ( 2 y 2 )( y)(+y) 2 y+y 2 y Simplifying comple fractions is basically just a combination of the concepts from the previous three lessons. The rational epressions in the numerator and/or denominator of the comple fraction need to be added or subtracted first (Lesson 8). Then the comple fraction gets converted to two rational epressions being divided, which we don t actually divide at all, we simply convert to multiplication by taking the reciprocal of the divisor (Lesson 8). When two fractions are being multiplied, we write numerator times numerator and denominator times denominator, but we don t actually multiply them because we need to cancel common factors (Lesson 7). Once we factor completely, we can simplify the rational epressions by canceling common factors (Lesson 7). Remember that the numerator and denominator of the rational epression need to be factored completely in order to simplify, and factoring we covered in Lessons 6 and 7. The ability to synthesize all the information from the previous three lessons is imperative to being able to simplify comple fractions.

2 It is imperative that you understand how to simplify, multiply, divide, add, and subtract rational epressions, as well as how to factor polynomials, in order to simplify comple fractions. All of those concepts are combined into one problem when simplifying comple fractions, and the inability to perform any of those tasks will prevent you from correctly answering these types of problems. Eample : Perform the indicated operations and simplify the epressions completely. a. b b. y 2 y 2 y 2 2

3 c. y 4y y 2 d. +( y )3 y + d. a

4 e. f. a + h h f. ( + h) 2 2 h

5 Negative Eponent Rule: - once again, to change the sign of an eponent, take the reciprocal of the factor or epression that has the negative eponent o 3 = = 5 3 (5) 2 = 25 2 Eample 2: Perform the indicated operations and simplify the following epression completely. Do not include negative eponents in your answer. (25) 25 ( ) ( ) ( ) ( 25 2) ( 25 2 )() (25)(25 2 ) ( 5)( + 5)() (25)()(5 )(5 + ) ( 5)( + 5) 25(5 )(5 + )

6 Re-arranging the terms or factors in an answer has no effect on that answer, so even if you epress the answer as (5+)( 5), the answer is 25(+5)(5 ) still the same and it still does not simplify any further. Also, keep in mind that in the original epression (25) 25 you cannot simply move (25) from the numerator to the denominator to change the sign of the eponent. Same with 25 ; you must take the reciprocal to change the sign of the eponent. Moving from numerator to denominator or vice versa is a shortcut that only works for factors, NOT terms. Eample 2: Perform the indicated operations and simplify the epressions completely. Do not include negative eponents in your answer. a. 3y 3 b. y (y) 2 y 2 2 b. a

7 c. ( 2 2 )( ) 2 d. ( )( 2 9) d. a

8 Answers to Eamples: a. ; b. 2 +y+y 2 2(+2) +y e. 2d. (+h) ; f. 3 2 (+3) ; 2+h 2 (+h) 2 ; 2a. ; c. 2y+ 4 ; d. 2 y+y 2 y 2 ; 3y +y ; 2b. y ; 2c. + 2 ( ) ;