Calc section 0.5

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Rational expressions

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Calc section 0.5

  1. 1. 0.5 Rational ExpressionsGoal: To Simplify and rationalize the denominator of rational expressions.<br />Rational Expression:<br />A rational expression is fraction whose numerator and denominator are polynomials<br />For example:<br />
  2. 2. Rational Functions:<br /><ul><li>Let u(x) and v(x) be polynomial functions. The function </li></ul>Is a rational function. The domain of this rational expression is the set of all real numbers for which v(x) ≠ 0.<br />
  3. 3. The key issue with the domain of rational expressions is that we can never divide by zero.<br />Recall: Dividing by zero is undefined<br />ex.<br />Domain = (-∞,0) U (0,∞)<br />
  4. 4. Example:<br />Domain = (-∞,3) U (3,7) U (7,∞)<br />Whenever we work with rational expressions, we have to make sure we check the domain. We never want to have an answer that results in the function becoming undefined.<br />
  5. 5. Find the domain<br /><ul><li>(-∞,2) U (2, ∞)</li></ul>Find the domain<br /><ul><li>(-∞,1) U (1, ∞)</li></li></ul><li>To simplify rational expressions:<br />Factor both numerator and denominator<br />Find domain of denominator<br />Reduce where possible<br /><ul><li>Example:</li></li></ul><li>(-∞, 3)U(3,7)U(7,∞) is the domain of the denominator<br />(-∞, 3)U(3,∞) We list this as our domain restriction since we have canceled the factor (x-3) on the graph there is a whole at x = 3<br />
  6. 6. Using your table, see what the value of y is when x = 3, what is different about x = 7? <br />*This will lead us to removable and non removable discontinuity in ch. 1<br />
  7. 7. <ul><li>Example: Simplify</li></ul>(-∞, 0)U(0,1)U(1,∞) is the domain of the denominator<br />x Є(-∞, 0)U(0,1)U(1,∞) <br />
  8. 8. <ul><li>To Multiply/Divide rational expressions:
  9. 9. Factor both numerator and denominator
  10. 10. Find domain of denominator for both expressions (also denominator of reciprocal when you divide)
  11. 11. Reduce where possible
  12. 12. Example:</li></li></ul><li>x Є(-∞, -3) U(-3,2)U(2,3)U(3,∞) <br />x Є(-∞,2)U(2,3)U(3,∞) <br />
  13. 13. EX.<br />x Є(-∞, -1) U(-1,0)U(0,1)U(1,∞) <br />x Є(-∞, -6) U(-6, -1) U(-1,0)U(0,1)U(1,∞) <br />
  14. 14. <ul><li>To Add/Subtract rational expressions:
  15. 15. Factor denominators
  16. 16. Find domain of denominator for both expressions
  17. 17. Reduce where possible
  18. 18. Find Common Denominator
  19. 19. Smiley Face
  20. 20. Other methods: What’s missing, etc.
  21. 21. Example:</li></li></ul><li>
  22. 22. <ul><li>Example:</li></li></ul><li><ul><li>Rationalizing Techniques:
  23. 23. Use the “conjugate”
  24. 24. Example:</li></li></ul><li><ul><li>Rationalizing The Numerator:
  25. 25. Example</li>

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