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Mastering Rational Expressions and Complex Fractions

Learn to simplify rational expressions and complex fractions by factoring, identifying undefined values, and simplifying using common factors. Understand how to simplify expressions and eliminate common factors.

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Mastering Rational Expressions and Complex Fractions

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  1. You factored polynomials. • Simplify rational expressions. • Simplify complex fractions. Then/Now

  2. rational expression • complex fraction Vocabulary

  3. A. Simplify . Eliminate common factors. ● Answer: Simplify a Rational Expression Look for common factors. Simplify. Example 1A

  4. B. Under what conditions is the expression undefined? Simplify a Rational Expression Just as with a fraction, a rational expression is undefined if the denominator equals zero. The original factored denominator is (y + 7)(y – 3)(y + 3). 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. Example 1B

  5. A. Simplify . A. B. C. D. Example 1A

  6. B. Under what conditions is the expression undefined? A. x = 4 or x = –4 B.x = –5 or x = 4 C.x = –5, x = 4, or x = –4 D.x = –5 Example 1B

  7. For what value(s) of p is undefined? A 5 B –3, 5 C 3, –5 D 5, 1, –3 Determine Undefined Values Read the Test Item You want to determine which values of p make the denominator equal to 0. Example 2

  8. Determine Undefined Values 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. p2 – 2p –15 = (p – 5)(p + 3) Factor the denominator. p – 5 = 0 or p + 3 = 0 Zero Product Property p = 5 p = –3 Solve each equation. Answer: B Example 2

  9. For what value(s) of p is undefined? A. –5, –3, –2 B. –5 C. 5 D. –5, –3 Example 2

  10. Simplify . Simplify Using –1 Factor the numerator and the denominator. b – 2 = –(–b + 2) or –1(2 – b) Simplify. Answer: –a Example 3

  11. Simplify . A.y – x B.y C.x D. –x Example 3

  12. Concept

  13. A.Simplify . Answer: Multiply and Divide Rational Expressions Simplify. Simplify. Example 4A

  14. B. Simplify Multiply by the reciprocal of the divisor. Simplify. Multiply and Divide Rational Expressions Example 4B

  15. Answer: Simplify. Multiply and Divide Rational Expressions Example 4B

  16. A. Simplify . A. B. C. D. Example 4A

  17. B. Simplify . A. AnsA B. AnsB C. AnsC D. AnsD Example 4B

  18. A.Simplify . Factor. Polynomials in the Numerator and Denominator 1 + k = k + 1,1 – k = –1(k – 1) = –1 Simplify. Answer: –1 Example 5A

  19. B.Simplify . Polynomials in the Numerator and Denominator Multiply by the reciprocal of the divisor. Factor. Example 5B

  20. Answer: Polynomials in the Numerator and Denominator Simplify. Example 5B

  21. A. Simplify . A. B. C.1 D.–1 Example 5A

  22. A. B. C. D. Example 5B

  23. Simplify . Express as a division expression. Multiply by the reciprocal of the divisor. Simplify Complex Fractions Example 6

  24. –1 Factor. Answer: Simplify Complex Fractions Simplify. Example 6

  25. Simplify . A. e B. C. e D. Example 6

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