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College Algebra Unit 8: Polynomial & rational functions

Polynomial and rational functions are the workhorses of algebra. Their zeros, end behavior and asymptotes tell you what the graph looks like.

Covers: End behavior, zeros & multiplicity, division, asymptotes

Study guide

Polynomial functions have end behavior set by the leading term. Even degree: both ends point the same way, up if the leading coefficient is positive and down if negative. Odd degree: the ends point opposite ways. A degree-n polynomial has at most n real zeros and at most n − 1 turning points.

Zeros come from factors. A factor (x − c)ᵏ gives a zero at c with multiplicity k: odd multiplicity crosses the x-axis and even multiplicity touches and turns. Use long division or synthetic division to divide polynomials. The Remainder Theorem says dividing f(x) by (x − c) leaves remainder f(c), and the Factor Theorem says f(c) = 0 exactly when (x − c) is a factor. The Rational Zero Theorem lists possible rational zeros as ±(factors of the constant)/(factors of the leading coefficient); test them, and divide out each zero you find.

Rational functions are ratios of polynomials. After factoring and canceling, a canceled factor leaves a hole, and a remaining denominator factor gives a vertical asymptote. Horizontal asymptotes compare degrees: a bottom degree bigger gives y = 0, equal degrees give y = ratio of leading coefficients, and a top degree bigger by one gives a slant asymptote found by division. Graphs can cross horizontal asymptotes but never vertical ones.

A polynomial of degree n has exactly n complex zeros, counting multiplicity. If its coefficients are real, nonreal zeros come in conjugate pairs: if 2 + i is a zero, so is 2 − i. If a polynomial's values at x = a and x = b have opposite signs, it has a real zero between a and b. Checking consecutive integers this way locates zeros.

To write a polynomial with given zeros r₁, r₂, …, use f(x) = a(x − r₁)(x − r₂)···, then find a from one more known point on the graph.

Key ideas

• End behavior of a polynomial depends on the leading term's degree and sign.
• A zero with odd multiplicity crosses the x-axis; even multiplicity touches and turns.
• Use long or synthetic division to divide polynomials; the Remainder Theorem says f(c) is the remainder when dividing by (x − c).
• Rational Zero Theorem: possible rational zeros are ±(factors of constant)/(factors of leading coefficient).
• Rational functions have vertical asymptotes where the reduced denominator is zero and holes where a factor cancels.

Formulas and rules

Common mistakes

• Calling a canceled factor a vertical asymptote. It's a hole.
• Assuming a graph can never cross a horizontal asymptote (it can, in the middle).

Words to know

End behavior

Set by the leading term: −2x⁵ rises left, falls right.

Even multiplicity zero

The graph touches the x-axis and turns around.

Remainder Theorem

Dividing f(x) by (x − c) leaves remainder f(c).

Vertical asymptote vs hole

Asymptote: denominator zero after reducing. Hole: a factor that cancels.

Horizontal asymptote when the bottom has higher degree

y = 0

Factor Theorem

f(c) = 0 exactly when (x − c) is a factor.

Turning points

A degree-n polynomial has at most n − 1.

Synthetic division

Shortcut for dividing by (x − c).

Rational Zero Theorem

Possible rational zeros: ±(factors of constant)/(factors of leading coefficient).

Hole

A canceled factor leaves a single missing point.

Slant asymptote

When the top's degree is one more than the bottom's; found by division.

Crossing a horizontal asymptote

Possible; graphs never cross vertical asymptotes.

Sample practice questions

27 more questions like these, from level 1 to 5, are in the practice set.