Precalculus · Unit 1: Polynomial and Rational Functions · 14 min read · Updated 2026-05-11
Rational functions and holes — AP Precalculus
AP Precalculus · Unit 1: Polynomial and Rational Functions · 14 min read
1. Core Concepts: What Is a Hole?★★☆☆☆⏱ 3 min
A rational function is defined as the ratio of two polynomials $f(x) = \frac{N(x)}{D(x)}$, where $N(x)$ is the numerator polynomial and $D(x)$ is a non-zero denominator polynomial. This topic is tested in both multiple-choice and free-response sections of the AP Precalculus exam, and falls within Unit 1, which makes up 27–31% of your total exam score.
2. Finding the x-coordinate of a Hole★★☆☆☆⏱ 4 min
To find the $x$-coordinate(s) of any hole, first fully factor both the numerator and denominator into linear factors. Any unique common linear factor of the form $(x-a)$ gives a hole at $x=a$. Roots of the denominator that are not roots of the numerator correspond to vertical asymptotes, not holes. Even if $(x-a)$ is raised to a power in both numerator and denominator, it only produces one hole at $x=a$.
3. Finding the y-coordinate of a Hole★★☆☆☆⏱ 3 min
Once you have the $x$-coordinate $a$ of a hole, the $y$-coordinate equals the value of the simplified function (after canceling all common factors) evaluated at $x=a$. This is equal to $\lim_{x \to a} f(x)$, the value the function approaches as $x$ nears $a$. Never plug $a$ into the original unsimplified function, as this will always give the indeterminate form $\frac{0}{0}$.
4. Distinguishing Holes from Vertical Asymptotes★★★☆☆⏱ 5 min
A very common AP exam question asks to classify all discontinuities of a given rational function. The classification rule is straightforward: after fully factoring the numerator and denominator, any root of the original denominator that is also a root of the numerator (i.e., corresponds to a common linear factor) is a hole. Any root of the original denominator that is not a root of the numerator is a vertical asymptote. On a graph, holes are marked as open circles, while vertical asymptotes are marked as dashed vertical lines.
Common Pitfalls
Why: Students forget that common factors must be canceled before evaluation, and confuse the indeterminate form $\frac{0}{0}$ with non-existence of the hole.
Why: Students confuse the fact that all discontinuities come from denominator roots with the type of discontinuity, forgetting that shared roots are removable.
Why: Students forget that non-linear common factors factor into multiple linear roots, each producing their own hole.
Why: Students confuse the existence of the limit at $x=a$ with the function being defined at $x=a$.
Why: Students think the exponent of the common factor creates multiple discontinuities at the same $x$-value.