---
title: "Multiplicity — AP Precalculus Definition & Exam Guide"
description: "Multiplicity is how many times a factor repeats in a polynomial. It decides whether a graph crosses or bounces at a zero, and whether a rational function has a hole or asymptote."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/multiplicity"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 1"
---

# Multiplicity — AP Precalculus Definition & Exam Guide

## Definition

In AP Precalculus, multiplicity is the number of times a linear factor (x − a) is repeated in a polynomial's factorization. Even multiplicity makes the graph touch and bounce off the x-axis at x = a; odd multiplicity makes it cross. Comparing multiplicities also determines holes vs. vertical asymptotes in rational functions.

## What It Is

Multiplicity counts repetition. If a polynomial factors as p(x) = (x − a)ⁿ · (other stuff), then the [zero](/ap-pre-calc/unit-1/polynomial-functions-complex-zeros/study-guide/Ex6Y5wBlobCpxdVr "fv-autolink") at x = a has multiplicity n. So p(x) = (x + 3)²(x)(x − 4)³ has a zero at −3 with multiplicity 2, at 0 with multiplicity 1, and at 4 with multiplicity 3. Per EK 1.5.A.2, a degree-n polynomial has exactly n complex zeros *when you count multiplicities*, which is why that example is a degree-6 polynomial even though it only has three distinct zeros.

Multiplicity isn't just bookkeeping. It controls graph behavior. At a zero with even multiplicity, the graph is [tangent](/ap-pre-calc/unit-3/tangent-function/study-guide/MmhWdpovNDRCpyBgCc0x "fv-autolink") to the x-axis (it touches and bounces). At a zero with odd multiplicity, the graph crosses. Then in Topics 1.9 and 1.10, multiplicity becomes the tiebreaker for rational functions. When the same value is a zero of both the numerator and denominator, you compare multiplicities to decide whether the graph has a hole or a [vertical asymptote](/ap-pre-calc/key-terms/vertical-asymptote "fv-autolink") at that input.

## Why It Matters

Multiplicity lives in [Unit 1](/ap-pre-calc/unit-1 "fv-autolink") (Polynomial and Rational Functions) and shows up in three separate topics. In Topic 1.5, it supports LO 1.5.A, where you identify key characteristics of a polynomial from its factorization, including whether the graph crosses or is tangent to the x-axis at each [real zero](/ap-pre-calc/key-terms/real-zero "fv-autolink"). In Topic 1.9 (LO 1.9.A), EK 1.9.A.1 says a vertical asymptote occurs at x = a when the multiplicity of a in the denominator is greater than its multiplicity in the numerator. In Topic 1.10 (LO 1.10.A), EK 1.10.A.1 flips that comparison. If the numerator's multiplicity is greater than or equal to the denominator's, the graph has a hole instead. One concept, three topics. Knowing it cold lets you read a factored function and predict its entire graph near each zero without plotting a single point.

## Connections

### [Real zero (Unit 1)](/ap-pre-calc/key-terms/real-zero)

Multiplicity is a property attached to a zero. EK 1.5.A.1 says a is a real zero of p exactly when (x − a) is a factor, and multiplicity just counts how many copies of that factor you get. No zero, no multiplicity to talk about.

### Removable discontinuity / holes (Unit 1)

A hole is what happens when the numerator 'wins' the multiplicity comparison. If a value's multiplicity in the numerator is at least its multiplicity in the denominator (EK 1.10.A.1), the factors cancel enough that the graph has a [removable discontinuity](/ap-pre-calc/key-terms/removable-discontinuity "fv-autolink") at that x-value instead of blowing up.

### [One-sided limit (Unit 1)](/ap-pre-calc/key-terms/one-sided-limit)

Multiplicity in the denominator controls [asymptote](/ap-pre-calc/key-terms/asymptote "fv-autolink") behavior on each side. Near a vertical asymptote, the denominator gets arbitrarily close to zero, and whether the leftover multiplicity is even or odd tells you if the graph shoots the same direction on both sides or opposite directions. One-sided limit notation is how you describe it.

### [Odd function (Unit 1)](/ap-pre-calc/key-terms/odd-function)

Don't merge these ideas, but they rhyme. Even vs. odd exponents drive both. An even exponent on a factor makes the graph bounce at that zero, just like an even power [function](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink") is symmetric over the y-axis. The parity of the exponent is doing the work in both cases.

## On the AP Exam

Multiplicity is classic Unit 1 multiple-choice territory, and it gets tested from two angles. Angle one is polynomial graphs. A question gives you zeros with multiplicities, like x = −3 (multiplicity 2), x = 0 (multiplicity 1), and x = 4 (multiplicity 3), and asks where the graph is tangent to the x-axis. The answer is wherever the multiplicity is even. Other stems flip it and ask which polynomial touches but doesn't cross at a given x-value. Angle two is rational functions. Given something like f(x) = (x² − 4)/(x − 2), you have to recognize that x = 2 is a zero of both numerator and denominator, compare multiplicities, and conclude there's a hole at x = 2 rather than a vertical asymptote. No released FRQ has used the word verbatim, but FRQs on polynomial and rational functions expect you to justify graph behavior at zeros, and multiplicity is the justification.

## multiplicity vs Degree

Degree describes the whole polynomial; multiplicity describes one zero. The degree is the sum of all the multiplicities. So (x + 3)²(x − 4)³ is a degree-5 polynomial with a multiplicity-2 zero at −3 and a multiplicity-3 zero at 4. If a question says 'degree 5 with three distinct real zeros,' multiplicity is how those facts coexist.

## Key Takeaways

- Multiplicity is the exponent on a linear factor (x − a) in the factored form of a polynomial, so (x − 5)³ means x = 5 is a zero with multiplicity 3.
- Even multiplicity means the graph touches the x-axis and bounces off (the graph is tangent there); odd multiplicity means the graph crosses.
- A degree-n polynomial has exactly n complex zeros when you count multiplicities, even if it has fewer distinct zeros.
- For rational functions, if the denominator's multiplicity at x = a is greater than the numerator's, there's a vertical asymptote at x = a.
- If the numerator's multiplicity at x = a is greater than or equal to the denominator's, the graph has a hole at x = a instead of an asymptote.
- When a value zeros out both the numerator and denominator, never assume asymptote or hole. Compare multiplicities first.

## FAQs

### What is multiplicity in AP Precalculus?

Multiplicity is the number of times a linear factor repeats in a polynomial's factorization. In p(x) = (x − 1)²(x + 4), the zero at x = 1 has multiplicity 2 and the zero at x = −4 has multiplicity 1. It's covered in Topics 1.5, 1.9, and 1.10 of Unit 1.

### Does a zero with even multiplicity cross the x-axis?

No. At a zero with even multiplicity, the graph touches the x-axis and bounces back, staying on the same side. The graph is tangent to the x-axis there. Only zeros with odd multiplicity actually cross.

### How do I know if a rational function has a hole or a vertical asymptote?

Compare the multiplicity of the zero in the numerator vs. the denominator. If the numerator's multiplicity is greater than or equal to the denominator's, it's a hole (EK 1.10.A.1). If the denominator's multiplicity is greater, it's a vertical asymptote (EK 1.9.A.1). For (x² − 4)/(x − 2), the factor (x − 2) appears once on top and once on bottom, so there's a hole at x = 2.

### What's the difference between multiplicity and degree?

Degree applies to the whole polynomial; multiplicity applies to one specific zero. Add up the multiplicities of all the zeros (counting complex ones) and you get the degree. A degree-6 polynomial might have just three distinct zeros if their multiplicities are 2, 1, and 3.

### Do complex zeros count toward multiplicity?

Yes. EK 1.5.A.2 says a polynomial of degree n has exactly n complex zeros when counting multiplicities. Complex zeros don't show up as x-intercepts on the graph, but they still count in the total.

## Related Study Guides

- [1.10 Rational Functions and Holes](/ap-pre-calc/unit-1/rational-functions-holes/study-guide/XgQqsfMcOkHxszGG)

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