Calc section 0.5
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Calc section 0.5

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Rational expressions

Rational expressions

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Calc section 0.5 Calc section 0.5 Presentation Transcript

  • 0.5 Rational ExpressionsGoal: To Simplify and rationalize the denominator of rational expressions.
    Rational Expression:
    A rational expression is fraction whose numerator and denominator are polynomials
    For example:
  • Rational Functions:
    • Let u(x) and v(x) be polynomial functions. The function
    Is a rational function. The domain of this rational expression is the set of all real numbers for which v(x) ≠ 0.
  • The key issue with the domain of rational expressions is that we can never divide by zero.
    Recall: Dividing by zero is undefined
    ex.
    Domain = (-∞,0) U (0,∞)
  • Example:
    Domain = (-∞,3) U (3,7) U (7,∞)
    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.
  • Find the domain
    • (-∞,2) U (2, ∞)
    Find the domain
    • (-∞,1) U (1, ∞)
  • To simplify rational expressions:
    Factor both numerator and denominator
    Find domain of denominator
    Reduce where possible
    • Example:
  • (-∞, 3)U(3,7)U(7,∞) is the domain of the denominator
    (-∞, 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
  • Using your table, see what the value of y is when x = 3, what is different about x = 7?
    *This will lead us to removable and non removable discontinuity in ch. 1
    • Example: Simplify
    (-∞, 0)U(0,1)U(1,∞) is the domain of the denominator
    x Є(-∞, 0)U(0,1)U(1,∞)
    • To Multiply/Divide rational expressions:
    • Factor both numerator and denominator
    • Find domain of denominator for both expressions (also denominator of reciprocal when you divide)
    • Reduce where possible
    • Example:
  • x Є(-∞, -3) U(-3,2)U(2,3)U(3,∞)
    x Є(-∞,2)U(2,3)U(3,∞)
  • EX.
    x Є(-∞, -1) U(-1,0)U(0,1)U(1,∞)
    x Є(-∞, -6) U(-6, -1) U(-1,0)U(0,1)U(1,∞)
    • To Add/Subtract rational expressions:
    • Factor denominators
    • Find domain of denominator for both expressions
    • Reduce where possible
    • Find Common Denominator
    • Smiley Face
    • Other methods: What’s missing, etc.
    • Example:
    • Example:
    • Rationalizing Techniques:
    • Use the “conjugate”
    • Example:
    • Rationalizing The Numerator:
    • Example