6.3. Complex Fractions

Transcription

1 6. Comple Fractions 1. Simplify comple fractions by simplifying the numerator and denominator (Method 1).. Simplify comple fractions by multiplying by a common denominator (Method ).. Compare the two methods of simplifying comple fractions. 4. Simplify rational epressions with negative eponents. Copyright 016, 01, 008 Pearson Education, Inc. 1

2 Comple Fraction A comple fraction is a quotient having a fraction in the numerator, denominator, or both. 4 m 1+, y,, and m+ 1 4 m y m 1 Copyright 016, 01, 008 Pearson Education, Inc.

3 Objective 1 Simplify comple fractions by simplifying the numerator and denominator (Method 1). Copyright 016, 01, 008 Pearson Education, Inc.

4 Simplifying a Comple Fraction (Method 1) Step 1 Simplify the numerator and denominator separately. Step Divide by multiplying the numerator by the reciprocal of the denominator. Step Simplify the resulting fraction if possible. Copyright 016, 01, 008 Pearson Education, Inc. 4

5 Eample 1 Simplifying Comple Fractions (Method 1) Use Method 1 to simplify each comple fraction. y + Both the numerator and the a. y denominator are already simplified. y Write as a division problem. y y+ y y y y+ y y y yy ( + ) yy ( ) Copyright 016, 01, 008 Pearson Education, Inc. 5

6 Eample 1 Use Method 1 to simplify each comple fraction. b Simplifying Comple Fractions (Method 1) (cont.) Simplify the numerator and denominator Copyright 016, 01, 008 Pearson Education, Inc. 6

7 Objective Simplify comple fractions by multiplying by a common denominator (Method ). Copyright 016, 01, 008 Pearson Education, Inc. 7

8 Simplifying a Comple Fraction (Method ) Step 1 Multiply the numerator and denominator of the comple fraction by the least common denominator of the fractions in the numerator and the fractions in the denominator of the comple fraction. Step Simplify the resulting fraction if possible. Copyright 016, 01, 008 Pearson Education, Inc. 8

9 Copyright 016, 01, 008 Pearson Education, Inc. 9 Use Method to simplify each comple fraction. a. Classroom Eample Simplifying Comple Fractions (Method ) The LCD is. Multiply the numerator and denominator by

10 Eample Simplifying Comple Fractions (Method ) (cont.) Use Method to simplify each comple fraction. 4 y + Multiply the numerator and b. y + 1 denominator by the LCD y(y + 1). y y 4 4 y + yy ( 1) y 1 + y[ yy ( + 1) ] + ( 1) + y 1 yy+ + y y( y 1) y + y[ yy ( + 1) ] yy ( 1) y + Copyright 016, 01, 008 Pearson Education, Inc. 10

11 Eample Simplifying Comple Fractions (Method ) (cont.) Use Method to simplify each comple fraction. 4 y[ yy ( + 1) ] + ( 1) y 1 yy+ + y[ yy ( + 1) ] yy ( 1) y + y ( y+ 1) + 4 y y ( y+ 1) ( y+ 1) y + y + 4y y + y y Copyright 016, 01, 008 Pearson Education, Inc. 11

12 Objective Compare the two methods of simplifying comple fractions. Copyright 016, 01, 008 Pearson Education, Inc. 1

13 Eample Simplifying Comple Fractions (Both Methods) Use both Method 1 and Method to simplify each comple fraction. a. Method y + y + y 4 ( y+ )( y) 5 y+ ( y+ )( y) 5 ( y+ )( y) y + 5( y ) Copyright 016, 01, 008 Pearson Education, Inc. 1

14 Eample Simplifying Comple Fractions (Both Methods) (cont.) Use both Method 1 and Method to simplify each comple fraction. a. Method 5 5 y + y + y 4 ( y )( y+ ) 5 ( y + )( y ) y + 5( y ) ( y+ )( y ) ( y )( y + ) Copyright 016, 01, 008 Pearson Education, Inc. 14

15 Eample Simplifying Comple Fractions (Both Methods) (cont.) Use both Method 1 and Method to simplify each comple fraction. b. Method 1 ba b a 1 1 a b 1 1 a b b ab b a b a ab a a b LCD ab LCD a b b a ab b a ab ab a b b a ab ab b a b a ab ab ( b a)( b + a) ab b + a Copyright 016, 01, 008 Pearson Education, Inc. 15

16 Eample Simplifying Comple Fractions (Both Methods) (cont.) Use both Method 1 and Method to simplify each comple fraction. 1 1 b. Method ab a b a b LCD a b ab 1 1 a b a b ab a b b a ab( b a) ( b+ a)( ba) ab b + a Copyright 016, 01, 008 Pearson Education, Inc. 16

17 Objective 4 Simplify rational epressions with negative eponents. Copyright 016, 01, 008 Pearson Education, Inc. 17

18 Eample 4 Simplifying Rational Epressions with Negative Eponents Simplify the epression, using only positive eponents in the answer. 1 a + b 1 LCD ab a 5b a b + ab + ab a b a b a b a b a b ab ab a b a b b + ab ab 5a Copyright 016, 01, 008 Pearson Education, Inc. 18

19 Eample 4 Simplifying Rational Epressions with Negative Eponents (cont.) Simplify the epression, using only positive eponents in the answer. 1 + y LCD y y+ 1 + y y + 1 y + y y y+ y+ y+ y 1 y + y y+ 1 y+ 1 y y 1 y + Copyright 016, 01, 008 Pearson Education, Inc. 19