CHAPTER 6 Polynomials: Factoring (continued) Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 6.1Multiplying and Simplifying Rational Expressions 6.2Division and Reciprocals 6.3Least Common Multiples and Denominators 6.4Adding Rational Expressions 6.5Subtracting Rational Expressions 6.6Solving Rational Equations 6.7Applications Using Rational Equations and Proportions

CHAPTER 6 Polynomials: Factoring Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 6.8Complex Rational Expressions 6.9Direct Variation and Inverse Variation

OBJECTIVES 6.1 Multiplying and Simplifying Rational Expressions Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aFind all numbers for which a rational expression is not defined. bMultiply a rational expression by 1, using an expression such as A/A. cSimplify rational expressions by factoring the numerator and the denominator and removing factors of 1. dMultiply rational expressions and simplify.

A rational expression is any expression that can be written as a quotient of two polynomials. Examples of rational expressions: Rational Expression 6.1 Multiplying and Simplifying Rational Expressions a Find all numbers for which a rational expression is not defined. Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Rational expressions are examples of algebraic fractions. They are also examples of fractional expressions. Because rational expressions indicate division, we must be careful to avoid denominators that are 0. Rational Expression 6.1 Multiplying and Simplifying Rational Expressions a Find all numbers for which a rational expression is not defined. Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution x 2  3x  28 = 0 (x  7)(x + 4) = 0 x  7 = 0 or x + 4 = 0 x = 7 or x =  4 Factoring Using the principle of zero products Solving each equation 6.1 Multiplying and Simplifying Rational Expressions a Find all numbers for which a rational expression is not defined. AFind all numbers for which the rational expression is undefined. Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

To multiply rational expressions, multiply numerators and multiply denominators: 6.1 Multiplying and Simplifying Rational Expressions Multiplying Rational Expressions Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Expressions that have the same value for all allowable (or meaningful) replacements are called equivalent expressions. 6.1 Multiplying and Simplifying Rational Expressions Equivalent Expressions Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution 6.1 Multiplying and Simplifying Rational Expressions b Multiply a rational expression by 1, using an expression such as A/A. BMultiply. Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. Using the identity property of 1. We arbitrarily choose 3x/3x as a symbol for 1.

Simplifying Rational Expressions A rational expression is said to be simplified when the numerator and the denominator have no factors (other than 1) in common. To simplify a rational expression, we first factor the numerator and denominator. We then identify factors common to the numerator and denominator, rewrite the expression as a product of two rational expressions (one of which is equal to 1), and then remove the factor equal to Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution Factoring the numerator and the denominator. Rewriting as a product of two rational expressions. Removing the factor 1 9x/9x = Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. C Simplify: Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) b)c) Solution a) 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. DSimplify: (continued) Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE b) c) 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. DSimplify: Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Canceling Canceling is a shortcut that can be used—and easily misused—to simplify rational expressions. Canceling must be done with care and understanding. Essentially, canceling streamlines the process of removing a factor equal to Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. ESimplify the expression by cancelling common factors. Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. FSimplify: (continued) Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution: We factor the numerator and denominator and look for common factors: 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. FSimplify: Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. GSimplify: (continued) Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Factoring 3  x =  x + 3 =  1(x  3) Rewriting as a product. Removing a factor equal to 1. Solution: 6.1 Multiplying and Simplifying Rational Expressions c Simplify rational expressions by factoring the numerator and the denominator and removing factors of 1. GSimplify: Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a ) b) Solution a) 6.1 Multiplying and Simplifying Rational Expressions d Multiply rational expressions and simplify. HMultiply and, if possible, simplify. (continued) Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE b) 6.1 Multiplying and Simplifying Rational Expressions d Multiply rational expressions and simplify. HMultiply and, if possible, simplify. Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE 6.1 Multiplying and Simplifying Rational Expressions d Multiply rational expressions and simplify. IMultiply and if possible, simplify. (continued) Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution 6.1 Multiplying and Simplifying Rational Expressions d Multiply rational expressions and simplify. IMultiply and if possible, simplify. Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.