Copyright © 2010 Pearson Education, Inc. All rights reserved Sec
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Rational Expressions and Functions Chapter 8
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Complex Fractions
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions Objectives 1. Simplify complex fractions by simplifying the numerator and denominator. (Method 1) 2. Simplify complex fractions by multiplying by a common denominator. (Method 2) 3. Compare the two methods of simplifying complex fractions. 4. Simplify rational expressions with negative exponents.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions Complex Fractions A complex fraction is an expression having a fraction in the numerator, denominator, or both. Examples of complex fractions include x 2 x 7 –, 5 m n, a 2 – 16 a + 2. a – 3 a 2 + 4
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions Simplifying Complex Fractions Simplifying a Complex Fraction: Method 1 Step 1 Simplify the numerator and denominator separately. Step 2 Divide by multiplying the numerator by the reciprocal of the denominator. Step 3 Simplify the resulting fraction, if possible.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 1 Simplifying Complex Fractions by Method 1 Use Method 1 to simplify each complex fraction. (a) d + 5 3d3d d – 1 6d6d = d + 5 3d3d d – 1 6d6d ÷= d + 5 3d3d 6d6d d – 1 · = 6 d ( d + 5) 3 d ( d – 1) = 2( d + 5) ( d – 1) Write as a division problem. Multiply by the reciprocal of. d – 1 6d6d Multiply. Simplify. Both the numerator and the denominator are already simplified, so divide by multiplying the numerator by the reciprocal of the denominator.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 1 Simplifying Complex Fractions by Method 1 Use Method 1 to simplify each complex fraction. Simplify the numerator and denominator. (Step 1) Write as a division problem. 3 x + 1 x Multiply by the reciprocal of. (Step 2) (b) 2 x 5 – 3 x 1 + = x 5 – x 2x2x x 1 + x 3x3x = x 2 x – 5 x 3 x + 1 ÷ = x 2 x – 5 x 3 x + 1 · = x 2 x – 5 3 x + 1 x = 2 x – 5 Multiply and simplify. (Step 3)
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions Simplifying Complex Fractions Simplifying a Complex Fraction: Method 2 Step 1 Multiply the numerator and denominator of the complex fraction by the least common denominator of the fractions in the numerator and the fractions in the denominator of the complex fraction. Step 2 Simplify the resulting fraction, if possible.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 2 Simplifying Complex Fractions by Method 2 Use Method 2 to simplify each complex fraction. Multiply the numerator and denominator by x since, = 1. (Step 1) Distributive property (a) 2 x 5 – 3 x 1 + Simplify. (Step 2) = · 1· 1 2 x 5 – 3 x 1 + = · x· x 2 x 5 – 3 x 1 + · x· x x x = 3 · x x 1 + · x· x 2 · x · x· x x 5 – = 3 x x – 5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 2 Simplifying Complex Fractions by Method 2 Use Method 2 to simplify each complex fraction. Multiply the numerator and denominator by the LCD, k ( k – 2). Distributive property Multiply; lowest terms (b) k – 2 4 3k3k + k k 7 – = 4 3k3k + · k ( k – 2) k k 7 – = k – · k ( k – 2) 3 k [ k ( k – 2) ] k 7 – · k ( k – 2) k [ k ( k – 2) ] = 3 k 2 ( k – 2) + 4 k k 2 ( k – 2) – 7( k – 2) = 3 k 3 – 6 k k k 3 – 2 k 2 – 7 k + 14
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 3 Simplifying Complex Fractions Using Both Methods Use both Method 1 and Method 2 to simplify each complex fraction. Method 1 (a) x – 1 3 x 2 – 1 4 = x – 1 3 ( x – 1)( x + 1) 4 = x – 1 3 ( x – 1)( x + 1) 4 ÷ = x – ( x – 1)( x + 1) · = 4 3( x + 1) (a) x – 1 3 x 2 – 1 4 = x – 1 3 · ( x – 1)( x + 1) ( x – 1)( x + 1) 4 · ( x – 1)( x + 1) = 4 3( x + 1) Method 2
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 3 Simplifying Complex Fractions Using Both Methods Use both Method 1 and Method 2 to simplify each complex fraction. Method 1 Method 2 (b) m 1 n 3 – mn n 2 – 9 m 2 = mnmn n mnmn 3m3m – mn n 2 – 9 m 2 = mn n – 3 m mn n 2 – 9 m 2 ÷ = mn n – 3 m ( n – 3 m )( n + 3 m ) mn · = n + 3 m 1 (b) m 1 n 3 – mn n 2 – 9 m 2 · mn = m 1 n 3 – mn n 2 – 9 m 2 · mn = n – 3 m n 2 – 9 m 2 = n – 3 m ( n – 3 m )( n + 3 m ) = n + 3 m 1
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Complex Fractions EXAMPLE 4 Simplifying Rational Expressions with Negative Exponents Simplify, using only positive exponents in the answer. b –1 c + bc –1 bc –1 – b –1 c b –1 c + bc –1 bc –1 – b –1 c = c b b c + b c c b – · bc c 2 + b 2 b 2 – c 2 = = b c c b – bc c b b c + Definition of negative exponent Distributive property Simplify using Method 2. (Step 1) Simplify. (Step 2)