Polynomial and Rational Functions Cheat Sheet

The core ideas of Polynomial and Rational Functions distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

A function of the form f(x) = a_n x^n + ... + a_1 x + a_0. The degree n determines end behavior and the maximum number of zeros.

If f(c)=0, then (x-c) is a factor. Zeros are where the graph crosses or touches the x-axis.

How many times a factor (x-c) appears. Odd multiplicity: graph crosses the axis. Even multiplicity: graph bounces off.

How f(x) behaves as x approaches positive or negative infinity. Determined by the leading term a_n x^n.

A ratio of two polynomials: f(x) = p(x)/q(x). Domain excludes values where q(x) = 0.

A vertical line x = a where the function approaches infinity. Occurs where the denominator is zero but the numerator is not.

The value y = L that f(x) approaches as x goes to infinity. Determined by comparing degrees of numerator and denominator.

Points where both numerator and denominator are zero due to a common factor. The factor cancels but the point is undefined.

Polynomial Function:A function of the form f(x)=a_n*x^n+...+a_0 where n is a non-negative integer.

Degree:The highest power of x in a polynomial. Determines end behavior and max number of zeros.

Leading Coefficient:The coefficient of the highest-degree term. Along with degree, determines end behavior.

Zero (Root):A value c where f(c)=0. Corresponds to an x-intercept on the graph.

Multiplicity:The number of times a factor (x-c) appears. Affects whether the graph crosses or bounces at that zero.

End Behavior:How f(x) behaves as x approaches positive or negative infinity.

Rational Function:A function that is a ratio of two polynomials: f(x) = p(x)/q(x).

Vertical Asymptote:A vertical line x=a where the function approaches infinity. Denominator is zero, numerator is not.

Horizontal Asymptote:A horizontal line y=L that the function approaches as x goes to infinity.

Hole:A removable discontinuity where both numerator and denominator are zero due to a common factor.

Slant Asymptote:An oblique asymptote that occurs when numerator degree exceeds denominator degree by 1.

Synthetic Division:A shortcut for dividing a polynomial by a linear factor (x-c).

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