1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Expressions and Functions: Multiplying and Dividing Rational Functions Multiplying Simplifying Rational Expressions and Functions Dividing and Simplifying 6.1

2. Slide 6- 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley An expression that consists of a polynomial divided by a nonzero polynomial is called a rational expression. The following are examples of rational expressions:

3. Slide 6- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Functions Like polynomials, certain rational expressions are used to describe functions. Such functions are called rational functions.

4. Slide 6- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The function given by Example gives the time, in hours, for two machines, working together to complete a job that the first machine could do alone in t hours and the other machine could do in 3t – 2 hours. How long will the two machines, working together, require for the job if the first machine alone would take (a) 2 hours? (b) 5 hours?

5. Slide 6- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution (a) (b)

6. Slide 6- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying The calculations that are preformed with rational expressions resemble those performed in arithmetic.

7. Slide 6- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Products of Rational Expressions To multiply two rational expressions, multiply numerators and multiply denominators: where

8. Slide 6- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply: Example Solution Multiplying the numerators and multiplying the denominators

9. Slide 6- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying Rational Expressions and Functions As in arithmetic, rational expressions are simplified by “removing” a factor equal to 1. For example, We removed the factor that equals 1.

10. Slide 6- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Because rational expressions often appear when we are writing functions, it is important that the function’s domain not be changed as a result of simplifying. For example, the domain of the function given by is assumed to be all real numbers for which the denominator is nonzero. Thus,

11. Slide 6- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In the previous example, we wrote F(x) in simplified form as There is a serious problem with stating that these are equivalent. The difficulty arises from the fact that, unless we specify otherwise, the domain of the function given by Thus the domain of G includes 5, but the domain of F does not. This problem can be addressed by specifying

12. Slide 6- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write the function given by Example Solution in simplified form. We first factor the numerator and denominator, looking for the largest factor common to both. Once the greatest common factor is found, we use it to write 1 and simplify:

13. Slide 6- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Note that the domain of g = {x | x  2/3 and x  -7} Factoring. The greatest common factor is (3x – 2). Rewriting as a product of two rational expressions. For x  2/3, we have (3x – 2)/(3x – 2) = 1. Solution (continued)

14. Slide 6- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution (continued) Removing the factor 1. To keep the same domain, we specify that x  2/3. Thus the simplified form is

15. Slide 6- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Canceling “Canceling” is a shortcut often used for removing a factor equal to 1 when working with fractions. Canceling removes factors equal to 1 in products. It cannot be done in sums or when adding expressions together. Simplifying the expression from the previous example might have been done faster as follows: When a factor that equals 1 is found, it is “canceled” as shown. Removing a factor equal to 1.

16. Slide 6- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Caution! Canceling is often performed incorrectly: Incorrect! Incorrect! Incorrect! In each situation, the expressions canceled are not both factors. Factors are parts of products. For example, 5 is not a factor of the numerator 5x – 2. If you can’t factor, you can’t cancel! When in doubt, don’t cancel! To check that these are not equivalent, substitute a number for x.

17. Slide 6- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Multiply. Then simplify by removing a factor equal to 1. Multiplying the numerators and also the denominators Factoring and finding common factors. Removing factors equal to 1. Simplifying

18. Slide 6- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Two expressions are reciprocals of each other if their product is 1. As in arithmetic to find the reciprocal of a rational expression, we interchange numerator and denominator. Dividing and Simplifying

19. Slide 6- 19 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quotients of Rational Expressions For any rational expressions A/B and C/D, with B, C, D  0, (To divide two rational expressions, multiply by the reciprocal of the divisor. We often say that we “invert and multiply.”)

20. Slide 6- 20 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Divide and, if possible, simplify. Multiplying by the reciprocal of the divisor Multiplying the numerators and denominators Factoring and removing factors equal to 1 Simplifying