Precalculus 2.5-2.7 Test Outline

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The Fundamental Theorem of Algebra

  • States that for any polynomial of degree > 0, there is at least one zero in the complex number system.
  • The linear factorization theorem states that a polynomial of degree n can be written with n linear factors whose zeros are in the complex number system.
  • This means that there are always n zeros for a polynomial of degree n, though some might be complex.
  • Complex zeros occur in conjugate pairs for polynomials with real coefficients.
  • Complex zero tricks:
    • quadratic formula to get complex zeros
    • creating complex conjugate zeros when given one of a pair
    • creating factors from complex zeros
    • multiplying factors from complex zeros to get quadratic
    • getting depressed polynomial with long division by quadratic
  • Factor polynomials to find zeros across the rationals, reals, and complex numbers. Tricks include:
    • common factor of x
    • rational roots
    • quadratic form
    • difference of squares
    • quadratic formula to find roots
  • writing polynomials with particular zeros (adding complex conjugate zeros as needed)

Rational Functions and Asymptotes

  • write statements about values of f(x) as x approaches an asymptote or infinity
  • notation for rational functions
  • find holes
  • find vertical asymptotes
  • find horizontal asymptotes
  • find domain

Graphs of Rational Functions

  • factor numerator and denominator
  • simplify to find holes
    • x coordinate from eliminated factor, same as in explicit domain
    • y coordinate by substituting into simplified function
  • remaining denominator zeros are vertical asymptotes
  • numerator zeros are also rational function zeros
  • y intercept is a0/b0 (constants)
  • horizontal asymptote
    • when n > m, none
    • when n = m, an/bm (leading coefficients)
    • when n < m, 0
  • draw curves between asymptotes through known points
    • may need to evaluate some more points
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