Polynomial, Rational, and Algebraic Functions
Core concepts
Polynomials are sums of terms aₙxⁿ. Key skills are factorising, using the factor and remainder theorems, sketching graphs from roots and their multiplicities, and identifying end behaviour from the leading term. A root of even multiplicity touches the axis; odd multiplicity crosses it. The remainder theorem says the remainder on dividing P(x) by (x − a) is P(a), and (x − a) is a factor exactly when P(a) = 0.
Rational functions introduce asymptotes. Vertical asymptotes occur where the denominator is zero but the numerator is not (a common factor instead gives a hole). Horizontal asymptotes depend on degrees: lower degree on top gives y = 0, equal degrees give the ratio of leading coefficients, higher degree on top gives no horizontal asymptote. Algebraic functions involving radicals require care with domain restrictions and extraneous solutions.
Traps to watch
- Cancelling a common factor and forgetting to record the resulting hole.
- Confusing a vertical asymptote with a point where the function is merely undefined but removable.
- Solving radical or rational equations and failing to check for extraneous roots.
- Misreading multiplicity when a factor appears as (x − 2)³ rather than (x − 2).
Rule of thumb
Factor completely first. Every cancelled factor is a hole; every remaining denominator zero is a vertical asymptote. Compare degrees to decide horizontal behaviour.
Worked example
For f(x) = (x² − 4)/(x² − x − 2), factor to get (x − 2)(x + 2) / [(x − 2)(x + 1)]. Cancel (x − 2): there is a hole at x = 2, a vertical asymptote at x = −1, and a horizontal asymptote y = 1 because the degrees are equal.