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Lecture complex fractions
1.
Section 6.3 Complex
Rational Expressions Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
2.
Objective #1 Simplify
complex rational expressions by multiplying by 1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 2
3.
Simplifying Complex Fractions
Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions. 5 x x 5 1 1 5 x Woe is me, for I am complex. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 3
4.
Complex Rational Expressions
Simplifying a Complex Rational Expression by Multiplying by 1 in the Form LCD T LCD 1) Find the LCD of all rational expressions within the complex rational expression. 2) Multiply both the numerator and the denominator of the complex rational expression by this LCD. 3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify each term. No fractional expressions should remain within the numerator and denominator of the main fraction. 4) If possible, factor and simplify. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4
5.
Simplifying Complex Fractions
x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 EXAMPLE Simplify: . 1 1 5 5 5 x x SOLUTION The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x. x 5 x x x x x x 1 1 5 5 5 5 1 1 5 5 5 Multiply the numerator and denominator by 5x.
6.
Simplifying Complex Fractions
Divide out common factors. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 x 1 x x x x x x 1 5 5 5 5 5 5 5 Use the distributive property. CONTINUED x 1 x x x x x x 1 5 5 5 5 5 5 5 2 25 5 x x Simplify. x 5 x 5 1 5 x Factor and simplify.
7.
Simplifying Complex Fractions
Simplify. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 CONTINUED 5 1 x Simplify. 5 x
8.
Simplifying Complex Fractions
1 x x x x x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 EXAMPLE Simplify: . 6 1 6 SOLUTION The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x. Multiply the numerator and denominator by (x + 6)x. 6 1 6 1 6 6 6 1 6 1 x x x x
9.
Simplifying Complex Fractions
Use the distributive property. CONTINUED 6 6 1 6 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 6 1 6 x x x x x x x x Divide out common factors. 6 6 1 6 6 1 6 x x x x x x x x Simplify. 6 x x 6 6 x x x x 6 36 6 x 2 x Simplify.
10.
Simplifying Complex Fractions
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10 CONTINUED 6 6x 2 36x Subtract. Factor and simplify. 6 1 6 6 x x 1 6 Simplify. x x
11.
x x x
y 1 1 1 1 1 1 xy y y x y y x y x y x y x y Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 1a. Simplify: 2 2 1 1 x y x y Multiply the numerator and denominator by the LCD of 2. y 2 2 2 2 2 2 2 2 2 2 2 2 y y y y y x y x x y y y y y 2 2 2 ( ) ( )( ) Objective #1: Example
12.
x x x
y 1 1 1 1 1 1 xy y y x y y x y x y x y x y Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 1a. Simplify: 2 2 1 1 x y x y Multiply the numerator and denominator by the LCD of 2. y 2 2 2 2 2 2 2 2 2 2 2 2 y y y y y x y x x y y y y y 2 2 2 ( ) ( )( ) Objective #1: Example
13.
Copyright ©
2013, 2009, 2006 Pearson Education, Inc. 13 Objective #1: Example 1b. Simplify: 1 1 7 7 x x Multiply the numerator and denominator by the LCD of x(x 7). 1 1 1 1 ( 7) ( 7) 7 ( 7) 7 7 7 ( 7) 7 7 ( 7) ( 7) 7 7 7 ( 7) 7 ( 7) 7 ( 7) 1 1 ( 7) ( 7) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
14.
Copyright ©
2013, 2009, 2006 Pearson Education, Inc. 14 Objective #1: Example 1b. Simplify: 1 1 7 7 x x Multiply the numerator and denominator by the LCD of x(x 7). 1 1 1 1 ( 7) ( 7) 7 ( 7) 7 7 7 ( 7) 7 7 ( 7) ( 7) 7 7 7 ( 7) 7 ( 7) 7 ( 7) 1 1 ( 7) ( 7) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
15.
Objective #2 Simplify
complex rational expressions by dividing. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15
16.
Simplifying Complex Fractions
Simplifying a Complex Rational Expression by Dividing 1) If necessary, add or subtract to get a single rational expression in the numerator. 2) If necessary, add or subtract to get a single rational expression in the denominator. 3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply. 4) If possible, simplify. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16
17.
Simplifying Complex Fractions
m 2 2 m m m 3 2 4 4 9 2 2 m 2 m m m m m 9 2 m m m m 2 2 3 3 m m 2 3 3 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 EXAMPLE Simplify: . 5 6 6 m m m m SOLUTION 1) Subtract to get a single rational expression in the numerator. 2 2 2 4 4 3 3 2 2 m m m 2 m m 2 2 2 2 3 3 2 2 3 3 3 3 2 m m m m m m m m m
18.
Simplifying Complex Fractions
3 2 CONTINUED m m m m 4 4 2 18 2 3 2 2 m m m 6 4 18 3 3 2 3 3 2 2 2) Add to get a single rational expression in the denominator. 3 m m 2 2 3 m m m m m 9 3 9 3 2 2 m m m m Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 m m m m m m 3 2 2 3 3 5 6 6 m m m m m m m m m 3 3 3 3 3 3 2 3 2 3 3 3 2 3 m m m m m m m m m 3 2 3 3 2 3 m m m m m m
19.
Simplifying Complex Fractions
3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify. m m m 6 4 18 m 3 m 3 m 2 9 m 3 2 3 2 m 2 2 m m m m 5 6 6 m m m 3 2 3 m m m 6 4 18 m m m 3 2 3 m m m 6 4 18 m m m 6 4 18 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 3 4 4 9 2 2 3 2 2 2 m m m m m m m 9 3 3 2 2 2 3 2 m m m m 9 3 3 2 2 2 3 2 m m m m 2 2 9 3 2 m m CONTINUED
20.
Objective #2: Example
x x x x x x x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20 2a. Simplify: 1 1 1 1 1 1 1 1 The LCD of the numerator is (x 1)(x 1). The LCD of the denominator is (x 1)(x 1).
21.
Objective #2: Example
x x x x x x x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21 2a. Simplify: 1 1 1 1 1 1 1 1 The LCD of the numerator is (x 1)(x 1). The LCD of the denominator is (x 1)(x 1).
22.
x x x
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 1 1 ( 1)( 1) ( 1)( 1) 2 1 2 1 1 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1) 1 1 ( 1)( 1) ( 1)( 1) 2 1 2 1 1 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1) x x x x x x x x 2 1 2 1 x x x 1)( 1) ( 1)( 1) x x x x x x x x x x x x x x x x x x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22 Objective #2: Example 2 2 2 2 2 2 2 1 ( 2 1) ( x 2 2 2 2 2 2 2 1 2 1 2 1 2 1 ( 1)( 1) ( 1)( 1) 4 ( 1)( 1) 4 ( 1)( 1) 4 2 x 2 2 ( x 1)( x 1) 2 x 2 2 2( x 2 1) ( x 1)( x 1) 2 x x 2 1 CONTINUED
23.
x x x
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 1 1 ( 1)( 1) ( 1)( 1) 2 1 2 1 1 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1) 1 1 ( 1)( 1) ( 1)( 1) 2 1 2 1 1 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1) x x x x x x x x 2 1 2 1 x x x 1)( 1) ( 1)( 1) x x x x x x x x x x x x x x x x x x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23 Objective #2: Example 2 2 2 2 2 2 2 1 ( 2 1) ( x 2 2 2 2 2 2 2 1 2 1 2 1 2 1 ( 1)( 1) ( 1)( 1) 4 ( 1)( 1) 4 ( 1)( 1) 4 2 x 2 2 ( x 1)( x 1) 2 x 2 2 2( x 2 1) ( x 1)( x 1) 2 x x 2 1 CONTINUED
24.
x 1
4 x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Objective #2: Example 2b. Simplify: 2 1 2 1 7 10 Rewrite the expression without negative exponents. Then multiply the numerator and denominator by the LCD of 2. x
25.
x 1
4 x x Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Objective #2: Example 2b. Simplify: 2 1 2 1 7 10 Rewrite the expression without negative exponents. Then multiply the numerator and denominator by the LCD of 2. x
26.
x x Copyright
© 2013, 2009, 2006 Pearson Education, Inc. 26 Objective #2: Example 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 1 1 4 1 7 10 7 10 1 4 4 1 1 7 10 7 10 1 1 4 ( 2)( 2) 2 7 10 ( 5)( 2) 5 x x x x x x x x x x x x x x x x x x x x x x x x x x CONTINUED
27.
x x Copyright
© 2013, 2009, 2006 Pearson Education, Inc. 27 Objective #2: Example 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 1 1 4 1 7 10 7 10 1 4 4 1 1 7 10 7 10 1 1 4 ( 2)( 2) 2 7 10 ( 5)( 2) 5 x x x x x x x x x x x x x x x x x x x x x x x x x x CONTINUED
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