Rational Expressions and Functions Chapter 8 Rational Expressions and Functions

8.3 Complex Fractions

8.3 Complex Fractions Objectives Simplify complex fractions by simplifying the numerator and denominator. (Method 1) Simplify complex fractions by multiplying by a common denominator. (Method 2) Compare the two methods of simplifying complex fractions. Simplify rational expressions with negative exponents. Copyright © 2010 Pearson Education, Inc. All rights reserved.

8.3 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 7 – , a2 – 16 a + 2 . a – 3 a2 + 4 5 m n , Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.

Simplifying Complex Fractions by Method 1 EXAMPLE 1 Simplifying Complex Fractions by Method 1 Use Method 1 to simplify each complex fraction. d + 5 3d d – 1 6d = d + 5 3d d – 1 6d ÷ (a) Write as a division problem. Both the numerator and the denominator are already simplified, so divide by multiplying the numerator by the reciprocal of the denominator. = d + 5 3d 6d d – 1 · Multiply by the reciprocal of . d – 1 6d = 6d(d + 5) 3d(d – 1) Multiply. = 2(d + 5) (d – 1) Simplify. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying Complex Fractions by Method 1 EXAMPLE 1 Simplifying Complex Fractions by Method 1 Use Method 1 to simplify each complex fraction. (b) 2 x 5 – 3 1 + = x 5 – 2x 1 + 3x = x 2x – 5 3x + 1 Simplify the numerator and denominator. (Step 1) ÷ = x 2x – 5 3x + 1 Write as a division problem. Multiply by the reciprocal of . (Step 2) · = x 2x – 5 3x + 1 3x + 1 x = 3x + 1 2x – 5 Multiply and simplify. (Step 3) Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.

Simplifying Complex Fractions by Method 2 EXAMPLE 2 Simplifying Complex Fractions by Method 2 Use Method 2 to simplify each complex fraction. (a) 2 x 5 – 3 1 + = · 1 2 x 5 – 3 1 + = · x 2 x 5 – 3 1 + Multiply the numerator and denominator by x since, = 1. (Step 1) x = 3 · x x 1 + · x 2 · x 5 – Distributive property = 3x + 1 2x – 5 Simplify. (Step 2) Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying Complex Fractions by Method 2 EXAMPLE 2 Simplifying Complex Fractions by Method 2 Use Method 2 to simplify each complex fraction. (b) k – 2 4 3k + k 7 – = k – 2 4 3k + · k(k – 2) k 7 – Multiply the numerator and denominator by the LCD, k(k – 2). = k – 2 4 + · k(k – 2) 3k [ k(k – 2) ] k 7 – k [ k(k – 2) ] Distributive property = 3k2(k – 2) + 4k k2(k – 2) – 7(k – 2) = 3k3 – 6k2 + 4k k3 – 2k2 – 7k + 14 Multiply; lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying Complex Fractions Using EXAMPLE 3 Simplifying Complex Fractions Using Both Methods Use both Method 1 and Method 2 to simplify each complex fraction. Method 1 Method 2 (a) x – 1 3 x2 – 1 4 = x – 1 3 (x – 1)(x + 1) 4 (a) x – 1 3 x2 – 1 4 = x – 1 3 (x – 1)(x + 1) 4 ÷ = x – 1 3 · (x – 1)(x + 1) (x – 1)(x + 1) 4 = x – 1 3 4 (x – 1)(x + 1) · = 4 3(x + 1) = 4 3(x + 1) Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying Complex Fractions Using 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 n2 – 9m2 = mn n 3m – n2 – 9m2 (b) m 1 n 3 – mn n2 – 9m2 · mn · mn = m 1 n 3 – mn n2 – 9m2 · mn = mn n – 3m n2 – 9m2 ÷ = mn n – 3m (n – 3m)(n + 3m) · = n – 3m n2 – 9m2 = n – 3m (n – 3m)(n + 3m) = n + 3m 1 = n + 3m 1 Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying Rational Expressions with 8.3 Complex Fractions EXAMPLE 4 Simplifying Rational Expressions with Negative Exponents Simplify , using only positive exponents in the answer. b –1c + bc –1 bc –1 – b –1c Definition of negative exponent = c b + – · bc b –1c + bc –1 bc –1 – b –1c Simplify using Method 2. (Step 1) · bc = b c – bc + Distributive property c2 + b2 b2 – c2 = Simplify. (Step 2) Copyright © 2010 Pearson Education, Inc. All rights reserved.