1. Chapter 9: Rational Expressions and Equations -Basically we are looking at expressions and equations where there is a variable in a denominator

2. 9.1 Multiplying and Dividing Rational Expressions Definitions and issues Simplifying Multiplying Dividing Complex Fractions

3. Definition A rational expression is a ratio of two polynomial expressions For example, (8 + x) / (13 + x) Generally, we can simplify rational expressions by cancelling out any factors common to the numerator and denominator Note that in the expression above, you CANNOT cancel the x terms.. You are NOT allowed to cancel terms that are WITHIN a sum or a difference To see why, suppose we had (3 + 5) / (3 + 8) This is equivalent to 8 / 11… BUT if we cancelled the 3’s, we would obtain 5 /8.. Which does NOT equal 8/11!!! TO SIMPLIFY A RATIONAL EXPRESSION, FACTOR THE NUMERATOR AND THE DENOMINATOR… THEN CANCEL ANY COMMON FACTORS

4. Issues Before you simplify a rational expression or combine rational expressions, you must look at the the denominator(s) and note any values which, when substituted in for a variable, would cause that denominator to equal zero These values are called EXCLUDED values You must find excluded values BEFORE you begin simplifying.. And list them along with your answer You may need to FACTOR a denominator to determine the excluded values… however, we usually need to factor the denominator ANYHOW See the examples on the next few slides Sometimes we ignore excluded values if there are multiple variables in the rational expression

5. Example 1-1a Simplify Look for common factors. 1 1 Factor. Simplify. Answer:

6. Example 1-1b Under what conditions is this expression undefined? A rational expression is undefined if the denominator equals zero. To find out when this expression is undefined, completely factor the denominator. Answer: The values that would make the denominator equal to 0 are –7, 3, and –3. So the expression is undefined at y = –7, y = 3, and y = –3. These values are called excluded values.

7. Example 1-1c a.Simplify b.Under what conditions is this expression undefined? Answer: Answer: undefined for x = –5, x = 4, x = –4

8. Example 1-2a Multiple-Choice Test Item For what values of p isundefined? A 5 B –3, 5 C 3, –5 D 5, 1, –3 Read the Test Item You want to determine which values of p make the denominator equal to 0.

9. Example 1-2b Solve the Test Item Look at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator. Factor the denominator. Solve each equation. Answer:B Zero Product Property or

10. Example 1-2c Multiple-Choice Test Item For what values of p isundefined? A –5, –3, –2 B –5 C 5 D –5, –3 Answer:D

11. Example 1-3a Simplify Factor the numerator and the denominator. Simplify. Answer:or –a or 1 1 a 1

12. Example 1-3b Simplify Answer: –x

13. Multiplying two rational expressions Factor the numerator AND denominator of each rational expression List the excluded values Cancel any factors common to the numerator and denominator Multiply the remaining factors in the numerator Multiply the remaining factors in the denominator One trick: sometimes it is advantage to factor a negative one (-1) from an expression, if it will allow you to cancel another factor out

14. Example 1-4a Simplify Simplify. Answer:Simplify. Factor. 1111111 1111111

15. Example 1-4b Simplify Factor. 1111111 1111111 1 Answer:Simplify.

16. Example 1-4c Simplify each expression. a. b. Answer:

17. Dividing Rational Expressions Recall that dividing by a fraction is the same as multiplying by the recipricol of that fraction Generally, it is advisable to rewrite a division problem as a multiplication problem before factoring and cancelling, etc.

18. Example 1-5a Simplify Answer:Simplify. Factor. 1111111 1111111 Multiply by the reciprocal of divisor.

19. Example 1-5b Simplify Answer:

20. Example 1-6a Simplify Multiply by the reciprocal of the divisor. 1 –11 1 11 Answer:Simplify.

21. Example 1-6b Simplify Multiply by the reciprocal of the divisor. Simplify.Answer: Factor. 1 11 1

22. Example 1-6c Answer: 1 Simplify each expression. a. b. Answer:

23. COMPLEX FRACTIONS A complex fraction is a rational expression whose numerator and/or denominator CONTAINS another rational expression! It’s kind of like a fraction within a fraction Just remember to treat this problem as a division problem – the numerator is being divided by the denominator

24. Example 1-7a Simplify Express as a division expression. Multiply by the reciprocal of divisor.

25. Example 1-7b Factor. 11–1 1 11 Simplify.Answer:

26. Example 1-7c Simplify Answer: