# Precalculus 2.5-2.7 Test Outline

From Huben's Wiki

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 a
_{0}/b_{0}(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