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Chapter 9: Rational Expressions and Equations

Chapter 9: Rational Expressions and Equations. Basically we are looking at expressions and equations where there is a variable in a denominator. 9.1 Multiplying and Dividing Rational Expressions. Definitions and issues Simplifying Multiplying Dividing Complex Fractions. Definition.

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Chapter 9: Rational Expressions and Equations

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

  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 calledexcluded values.

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

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

  9. Factor the denominator. or Zero Product Property Solve each equation. 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 solutionand one negative solution. Therefore, you can eliminate choices A and D.Factor thedenominator. Answer: B

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

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

  12. Simplify Example 1-3b 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. Simplify 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Factor. Simplify. Answer: Simplify. Example 1-4a

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

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

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

  19. Simplify Answer: Example 1-5b

  20. Simplify Multiply bythe reciprocalof the divisor. 1 –1 1 1 1 1 Answer: Simplify. Example 1-6a

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

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

  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. Simplify Express as adivision expression. Multiply by thereciprocal of divisor. Example 1-7a

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

  26. Simplify Answer: Example 1-7c

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