Root Finder with Multiplicity

Math Algebra • Polynomial and Rational Functions

Written by STEM Calculators Team Published December 14, 2025 Updated February 24, 2026

Enter a polynomial in x. Use ^ for powers and parentheses. Implicit multiplication is allowed: 2x(x-1), (x-1)(x+1), x^3-6x^2+11x-6. Exponents must be nonnegative integers.

Polynomial \(P(x)\): Mode:

Graph window

\(x_{\min}\): \(x_{\max}\): Resolution (points):

Numerical settings

Root tolerance: Max iterations: Multiplicity clustering: Real threshold \(|\Im|\le\):

Zoom to a root

Zoom width (\(\pm\)): Select root:

Options:

Ready

Enter \(P(x)\), then press Calculate.

Drag to pan. Mouse wheel zooms (cursor-centered). Double-click resets view. Axis numbers are shown.

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2. Root Finder With Multiplicity — Theory

A root (or zero) of a polynomial \(P(x)\) is a value \(r\) such that \(P(r)=0\). This tool finds roots (real and complex) and estimates each root’s algebraic multiplicity.

1) What “multiplicity” means

\[ r \text{ is a root of multiplicity } m \quad\Longleftrightarrow\quad (x-r)^m \text{ divides } P(x)\text{, but }(x-r)^{m+1}\text{ does not.} \]

If \(m=1\), the graph usually crosses the \(x\)-axis at \(r\).
If \(m=2,4,\dots\) (even), the graph typically touches the axis and turns around.
Larger \(m\) makes the graph flatter near the root.

2) A derivative test for repeated roots

A classic fact: if \(r\) is a multiple root, then \(r\) is also a root of the derivative.

\[ m\ge 2 \quad\Rightarrow\quad P(r)=0 \text{ and } P'(r)=0. \]

More strongly, \(r\) has multiplicity \(m\) if \(P(r)=P'(r)=\cdots=P^{(m-1)}(r)=0\) but \(P^{(m)}(r)\ne 0\).

3) Rational Root Theorem (quick candidate list)

If \(P(x)=a_nx^n+\cdots+a_0\) has integer coefficients, any rational root \(r=\dfrac{p}{q}\) in lowest terms must satisfy:

\[ p\mid a_0,\qquad q\mid a_n. \]

So you can test a finite set of candidates \(\pm\dfrac{p}{q}\). The calculator can list these candidates (when coefficients are integers).

4) Complex roots and the Fundamental Theorem of Algebra

Every nonconstant polynomial of degree \(n\) has exactly \(n\) complex roots when counted with multiplicity. Complex roots of real-coefficient polynomials occur in conjugate pairs:

\[ (a+bi)\text{ is a root } \Rightarrow (a-bi)\text{ is also a root.} \]

5) What the calculator is doing

Smart mode: tries to extract obvious integer roots by testing divisors of the constant term and deflating \((x-r)\) repeatedly to get multiplicity.
Numeric mode: computes all complex roots numerically and estimates multiplicities by clustering nearby roots.
Graph: plots \(y=P(x)\) and labels real roots \((r,0)\), using a bigger marker for multiplicity \(\ge 2\).

6) Tolerances (why they matter)

Numerical root-finding returns approximations. Two important settings:

Real threshold: treats a root as real if \(|\Im(z)|\) is small enough.
Multiplicity clustering: groups very close computed roots together to estimate multiplicity.

If a repeated root “splits” into nearby values, increase clustering slightly. If distinct roots get merged, decrease it.

Frequently Asked Questions

What does multiplicity mean for a polynomial root?

A root r has multiplicity m if (x-r)^m divides P(x) but (x-r)^(m+1) does not. Higher multiplicity typically makes the graph flatter near r, and even multiplicity often corresponds to touching the x-axis instead of crossing.

How does the calculator decide whether a root is real or complex?

A computed root is treated as real when its imaginary part is small enough to satisfy the real-threshold setting, i.e., |Im(z)| is below the cutoff. Otherwise it is reported as complex.

Why can repeated roots appear as several nearby roots numerically?

Finite-precision computations can split a multiple root into a cluster of very close approximations. The multiplicity clustering setting groups nearby roots together to estimate the multiplicity.

What is the Rational Root Theorem list used for?

If P(x) has integer coefficients, any rational root p/q (in lowest terms) must have p dividing the constant term and q dividing the leading coefficient. The candidate list provides the finite set of values to test for exact rational roots.

How many roots should a polynomial have when multiplicity is counted?

A nonconstant polynomial of degree n has exactly n complex roots when counted with multiplicity. For real-coefficient polynomials, nonreal complex roots occur in conjugate pairs.

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