Root of a Polynomial¶

Definition¶

Let $p\in\mathbb{F}[x]$ be a polynomial with coefficients in a field $\mathbb{F}$. A scalar $\lambda\in\mathbb{F}$ is a root or zero of $p$ if

Interpretation¶

A root is an input at which the polynomial evaluates to zero: it is the zero of a function when that function is the polynomial $p$. The defining test is evaluation at the proposed root, not a zero coefficient of $p$.

The graph of $p(x)=\tfrac{1}{4}(x+2)(x-1)(x-3)$ meets the $x$-axis exactly at the red points $-2$, $1$ and $3$. Each root comes from a factor: at $x=1$ the factor $x-1$ is zero, so the whole product is zero. Conversely, polynomial long division shows that $\lambda$ is a root if and only if $x-\lambda$ divides $p(x)$, so finding roots and splitting off linear factors are the same task. Since each root splits off a factor of degree one, a nonzero polynomial of degree $n$ has at most $n$ roots.

The field in the definition matters. The polynomial $x^2+1$ has no root in $\mathbb{R}$, since $x^2+1\ge1$ for every real $x$, but over $\mathbb{C}$ it has the roots $\mathrm{i}$ and $-\mathrm{i}$, because $\mathrm{i}^2=-1$$-1$. By the fundamental theorem of algebra, every nonconstant polynomial with complex coefficients has a root in $\mathbb{C}$, which is one reason why the complex numbers are convenient scalars in linear algebra.

This returns later in the course: the eigenvalues of a square matrix are the roots of its characteristic polynomial.

Key Results¶

• A field is algebraically closed if every nonconstant polynomial over it has a root in that field.
• Dividing $p$ by a nonzero polynomial $d$ gives unique $q,r$ with $p=qd+r$ and either $r=0$ or $\deg r<\deg d$$\mathrm{deg}d$. In particular, $\lambda$ is a root of $p$ if and only if $(x-\lambda)$ divides $p$; the multiplicity of the root $\lambda$ is the largest $m$ such that $(x-\lambda)^m$ divides $p$.

Examples¶

• $2$ is a root of $p(x)=x-2$$2$.
• For $p(x)=x^2-1$$1$, the roots are $1$ and $-1$.

Prerequisites¶

• Polynomial
• Zero of a Function

Related Concepts¶

• Characteristic Polynomial
• Eigenvalue-Root Theorem
• Polynomial Long Division

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: zero of a polynomial, polynomial zero, polynomial root Symbols: complex numbers, element of, lambda, real numbers

Last updated: 20 October 2026 03:02