Polynomial and Rational Functions: Zeros, End Behavior, and Asymptotes
What you must understand
For polynomials: degree, leading coefficient, end behavior, multiplicity of zeros, and the relationship between factors and x-intercepts. For rational functions: domain exclusions, vertical asymptotes, holes, horizontal or slant asymptotes, and sign analysis. You should connect factored form to graphical features and use the Fundamental Theorem of Algebra to count complex roots.
Traps that show up
- Multiplicity changes the graph's shape: odd multiplicity crosses the axis; even multiplicity touches and turns. Multiplicity 3 flattens at the intercept.
- Holes versus vertical asymptotes: a common factor in numerator and denominator cancels to create a hole, not an asymptote. Check whether the canceled factor truly vanishes at that x-value.
- Horizontal asymptote rules depend on degree comparison: numerator degree < denominator → y = 0; equal → ratio of leading coefficients; numerator one degree higher → slant asymptote; more than one degree higher → no horizontal or slant asymptote.
- End behavior: determined solely by the leading term; even degree with positive leading coefficient rises on both ends.
Rule of thumb
Factor completely first. Cancel common factors to locate holes. Then compare remaining degrees for asymptote behavior, and use sign charts to determine where the graph lies above or below the x-axis.
Worked example
f(x) = (x² − 4)/(x² − x − 2) = (x−2)(x+2)/[(x−2)(x+1)]. Cancel (x−2): a hole at x = 2. Vertical asymptote at x = −1. Degrees equal, so horizontal asymptote y = 1.