Multiplicity of Roots

Input Polynomial

Factored Polynomial

Use ^ or ² for exponents. Supported: (x−r)^n, (x+r), (x)²

Examples

Standard Form Polynomial

Enter coefficients in descending order. Uses Rational Root Theorem + synthetic division.

Examples

Root Analysis

Enter a polynomial and press Analyze Roots to see root multiplicities and graph behavior.

Zero	Multiplicity	Graph Behavior

Multiplicity & Graph Behavior

n=1

Crosses the x-axis

Odd multiplicity (1, 3, 5…): graph passes straight through the x-axis at the root.
n=2

Touches & bounces back

Even multiplicity (2, 4…): graph touches the x-axis and reverses direction — like a ball bouncing.
n=3

Flattens, then crosses

Higher odd multiplicities: graph is very flat near the root but still crosses. The curve looks S-shaped near the intercept.
n=4

Very flat bounce

Higher even multiplicities: graph barely grazes the x-axis and bounces back, appearing almost tangent.

Higher multiplicity = flatter curve near that root. The graph is tangent to the x-axis for even multiplicity.

The Degree Verification Rule

degree = sum of all multiplicities

If p(x) = a(x−r₁)ⁿ¹(x−r₂)ⁿ²⋯(x−rₖ)ⁿᵏ, then the degree of the polynomial equals n₁ + n₂ + ⋯ + nₖ.

Use this to verify you have found ALL roots. If the sum of multiplicities is less than the degree, there are still roots to find (possibly complex or irrational).

Factor completely first, then read off each power.
A root with even multiplicity does NOT cross the x-axis.
Count multiplicity when listing zeros on a test.
Complex roots always come in conjugate pairs.

Polynomials still tripping you up?