Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 7 Rational Expressions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 7.1 Rational Functions and Multiplying and Dividing Rational Expressions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Rational expression is an expression that can be written as the quotient of two polynomials P and Q as long as Q is not 0. Examples of Rational Expressions

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall A rational expression is undefined if the denominator is 0. If a variable in a rational expression is replaced with a number that makes the denominator 0, we say that the rational expression is undefined for this value of the variable.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the domain of the rational expression. Solution The domain of each function will contain all real numbers except those values that make the denominator 0. No matter what the value of x, the denominator is never 0. The domain is all real numbers.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the domain of the rational expression. Solution To find the values of x that make the denominator 0, we solve the equation “denominator = 0”: The domain must exclude 2. The domain of g is all real numbers except 2. Example

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the domain of the rational expression. Solution Set the denominator equal to 0. The domain of h is all real numbers except 2 and  3. Example

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall To simplify a rational expression, or to write it in lowest terms, we use a method similar to simplifying a fraction. Fundamental Principle of Rational Expressions For any rational expression and any polynomial R, where R ≠ 0, or, simply,

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplifying or Writing a Rational Expression in Lowest Terms 1. Completely factor the numerator and denominator of the rational expression. 2. Divide out factors common to the numerator and denominator. (This is the same thing as “removing the factor of 1.”)

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify the rational expression. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify the rational expression. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify the rational expression. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Simplify the rational expression. Solution The terms in the numerator differ by the sign of the terms in the denominator. The polynomials are opposites of each other. Factor out a  1 from the numerator.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplify Solution Example

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Simplify Solution Example

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Multiplying Rational Expressions The rule for multiplying rational expressions is To multiply rational expressions, you may use these steps: 1. Completely factor each numerator and denominator. 2. Use the rule above and multiply the numerators and denominators. 3. Simplify the product by dividing the numerator and denominator by their common factors. as long as Q  0 and S  0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Multiply. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Multiply. Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Dividing Rational Expressions The rule for dividing rational expressions is To divide by a rational expression, use the rule above and multiply by its reciprocal. Then simplify if possible. as long as Q  0, S  0 and R  0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Perform the indicted operation. Solution