Precalculus 2.5-2.7 Test Outline

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, a_n/b_m (leading coefficients)
  • when n < m, 0
• draw curves between asymptotes through known points
  • may need to evaluate some more points