1. Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

2. 6.1 Rational Expressions and Functions: Multiplying and Dividing ■ Rational Functions ■ Simplifying Rational Expressions and Functions ■ Multiplying and Simplifying ■ Dividing and Simplifying ■ Vertical Asymptotes

3. Slide 6- 3 Copyright © 2012 Pearson Education, Inc. Rational Expression A rational expression is any expression that consists of a polynomial divided by a nonzero polynomial. Examples of rational expressions:

4. Slide 6- 4 Copyright © 2012 Pearson Education, Inc. Rational Functions Like polynomials, certain rational expressions are used to describe functions. Such functions are called rational functions. Graphs from rational functions vary widely in shape, but the following are some general statements that can be made. 1.Graphs of rational functions may not be continuous, that is there may be a break in the graph. 2.The domain of a rational function may not include all real numbers.

5. Slide 6- 5 Copyright © 2012 Pearson Education, Inc. Simplifying Rational Expressions and Functions 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 1.

6. Slide 6- 6 Copyright © 2012 Pearson Education, Inc. Example Simplify:. Solution Factoring the numerator and the denominator. Note the common factor of 9x. Rewriting as a product of two rational expressions. Removing the factor 1 9x/9x = 1

7. Slide 6- 7 Copyright © 2012 Pearson Education, Inc. Example Simplify: a) b) c) Solution a)

8. Slide 6- 8 Copyright © 2012 Pearson Education, Inc. Example continued b)c)

9. Slide 6- 9 Copyright © 2012 Pearson Education, Inc. Canceling Canceling is a shortcut that can be used—and easily misused—when we are working with rational expressions. Canceling must be done with care and understanding. Essentially, canceling streamlines the process of removing a factor equal to 1. Example

10. Slide 6- 10 Copyright © 2012 Pearson Education, Inc. Example Simplify: Solution We factor the numerator and denominator and look for common factors:

11. Slide 6- 11 Copyright © 2012 Pearson Education, Inc. The Product of Two Rational Expressions To multiply rational expressions, multiply numerators and multiply denominators: where B ≠ 0, D ≠ 0. Then factor, and if possible, simplify the result. Multiplication

12. Slide 6- 12 Copyright © 2012 Pearson Education, Inc. Example Multiply. Then simplify by removing a factor of 1. a) b) Solution a)

13. Slide 6- 13 Copyright © 2012 Pearson Education, Inc. Example continued b)

14. Slide 6- 14 Copyright © 2012 Pearson Education, Inc. Example Multiply and if possible, simplify. Solution

15. Slide 6- 15 Copyright © 2012 Pearson Education, Inc. Division Quotients of Two Rational Expressions For any rational expressions A/B and C/D, with B, C, D ≠ 0, Then factor and, if possible, simplify.

16. Slide 6- 16 Copyright © 2012 Pearson Education, Inc. Example Divide: a) b) Solution a) b) Multiplying by the reciprocal of the divisor Multiplying rational expressions

17. Slide 6- 17 Copyright © 2012 Pearson Education, Inc. Example Divide and, if possible, simplify: Solution

18. Slide 6- 18 Copyright © 2012 Pearson Education, Inc. Example Divide and, if possible, simplify: Solution

19. Slide 6- 19 Copyright © 2012 Pearson Education, Inc. Example Divide and, if possible, simplify: Solution

20. Slide 6- 20 Copyright © 2012 Pearson Education, Inc. Vertical Asymptotes If a function f(x) is described by a simplified rational expression, and a is a number that makes the denominator 0, then x = a is a vertical asymptote of the graph f(x).

21. Slide 6- 21 Copyright © 2012 Pearson Education, Inc. Example Determine the vertical asymptotes of the graph of Solution First simplify the rational expression describing the function.

22. Slide 6- 22 Copyright © 2012 Pearson Education, Inc. continued The denominator of the simplified expression x + 4, is 0 when x = –4. Thus x = –4 is the vertical asymptote of the graph.