Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 7 Rational Expressions and Equations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Rational Expressions and Equations 7.1Simplifying, Multiplying, and Dividing Rational Expressions 7.2Adding and Subtracting Rational Expressions 7.3Simplifying Complex Rational Expressions 7.4Solving Equations Containing Rational Expressions CHAPTER 7

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Simplifying, Multiplying, and Dividing Rational Expressions 1.Simplify rational expressions. 2.Multiply rational expressions. 3.Divide rational expressions. 4.Evaluate rational functions. 5.Find the domain of a rational function. 7.1

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Rational expression: An expression that can be written in the form, where P and Q are polynomials and Q 0. Some rational expressions are

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Simplifying Rational Expressions to Lowest Terms To simplify a rational expression to lowest terms, 1. Factor the numerator and denominator completely. 2. Divide out all common factors in the numerator and denominator. 3.Multiply the remaining factors in the numerator and the remaining factors in the denominator. Fundamental Principle of Rational Expressions

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Example Simplify Solution Write the numerator and denominator in factored form, then eliminate the common factors. Multiply the remaining factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Example Simplify. Solution Write the numerator and denominator in factored form, then divide out the common factors, x and x + 2.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Simplify. Solution Factor the numerator and denominator completely. Then divide out the common factor, y – 5.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Simplify. Solution Factor out the GCF. Factor the polynomial factors. Divide out common factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Simplify. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 Example Simplify. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Multiplying Rational Expressions Sign Placement

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Multiplying Rational Expressions To multiply rational expressions, 1. Factor each numerator and denominator completely. 2.Divide out factors common to both the numerator and the denominator. 3. Multiply numerator by numerator and denominator by denominator. 4. Simplify as needed.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Multiply. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Solution Write numerators and denominators in factored form. Multiply the remaining numerator factors and denominator factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Example Multiply. Solution Write numerators and denominators in factored form.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Example Multiply. Solution Write numerators and denominators in factored form. Multiply the remaining numerator factors and denominator factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 Dividing Rational Expressions To divide rational expressions 1. Write an equivalent multiplication statement using 2. Simplify using the procedure for multiplying rational expressions.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19 Example Divide. Solution Write an equivalent multiplication statement. Divide out common factors, and multiply remaining factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20 Example Divide. Solution Write an equivalent multiplication statement. Divide out common factors, and multiply remaining factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 21 Example Divide. Solution Write an equivalent multiplication statement. Divide out common factors, and multiply remaining factors.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 22 Rational function: A function expressed in terms of rational expressions.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 23 Example Given find f(3). Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 24 Finding the Domain of a Rational Function To find the domain of a rational function 1. Write an equation that sets the denominator equal to Solve the equation. 3. Exclude the value(s) found in step 2 from the function’s domain.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 25 Example Find the domain of Solution The function is undefined if y is replaced by 0, –4, or –1, so the domain is {x|x  0,  1,  4}. Set the denominator equal to 0. Factor out the monomial GCF, y. Use the zero factor theorem.