Roots of a Polynomial

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Definition

Let $p ( x )$ be a polynomial with coefficients in a field $\mathbb{F}$, typically $\mathbb{R}$ or $\mathbb{C}$. A root or zero of $p$ is any element $r \in \mathbb{F}$ such that

More precisely, if we have the polynomial:

with $a_{n} \neq 0$, then $r$ is a root. Substituting $x = r$ gives a linear combination of the coefficients that equals zero. The terms root and zero are used interchangeably.

For a polynomial $p : \mathbb{R} \rightarrow \mathbb{R}$, the real roots are the $x$-intercepts of its graph. The multiplicity of a root affects the graph locally. At a simple root (multiplicity one), the graph crosses the $x$-axis cleanly and is not tangent.

For a root with even multiplicity, the graph touches the $x$-axis but does not cross it. Since $( x - r )^{m} \geq 0$ for even $m$, the polynomial does not change sign at $r$, and the graph bounces back to the same side of the axis.

For roots with odd multiplicity greater than one ($m \geq 3$), the graph crosses the axis but appears flatter at the intercept, with this flattening becoming more pronounced as multiplicity increases, giving an inflexion-like appearance.

These properties follow from the local factorization:

with $q ( r ) \neq 0$. Since $q$ is continuous and nonzero at $r$, it maintains a constant sign in some neighborhood of $r$, so the sign of $p ( x )$ near $r$ is determined entirely by the factor $( x - r )^{m}$.

• When $m$ is odd, $( x - r )^{m}$ changes sign as $x$ passes through $r$, so $p$ crosses the axis.
• When $m$ is even, $( x - r )^{m} \geq 0$ on both sides of $r$, so $p$ does not change sign and the graph returns to the same side of the axis.

A non-zero polynomial of degree $n$ over any field has at most $n$ roots, counted with multiplicity. This follows from the fact that a polynomial of degree $n$ cannot be divisible by more than $n$ linear factors.

In particular, two distinct polynomials of degree at most $n$ cannot agree at more than $n$ points. If $p ( x ) - q ( x )$ has degree at most $n$ and vanishes at $n + 1$ points, then $p \equiv q$.

Rational root theorem

Given a polynomial with integer coefficients:

the rational root theorem identifies a finite set of candidates for rational roots. If $r = s / q$ in lowest terms, with $s , q \in \mathbb{Z}$ and $q > 0$, is a root of $p ( x )$, then necessarily $s \mid a_{0}$ and $q \mid a_{n}$.

This reduces the search for rational roots to a finite collection of fractions, each of which can be verified by direct substitution or synthetic division.

The Fundamental Theorem of Algebra

In the field of complex numbers $\mathbb{C}$, every non-constant polynomial has at least one root. Applying the factor theorem repeatedly, any polynomial of degree $n \geq 1$ decomposes completely into linear factors over $\mathbb{C}$:

where $m_{1} + m_{2} + \hdots + m_{k} = n$. Counting roots with their multiplicities, a degree-$n$ polynomial has exactly $n$ roots in $\mathbb{C}$. This property characterises $\mathbb{C}$ as an algebraically closed field.

Over $\mathbb{R}$, the complex roots of a real polynomial occur in conjugate pairs. If $r = \alpha + \beta i$ with $\beta \neq 0$ is a root of $p \in \mathbb{R} [ x ]$, then $\bar{r} = \alpha - \beta i$ is also a root, and the two factors combine into an irreducible quadratic over $\mathbb{R}$:

Consequently, every real polynomial of odd degree has at least one real root. The factored form also establishes a direct relationship between roots and coefficients. Expanding we have:

Comparing with:

this yields Vieta’s formulas, which express each coefficient as an elementary symmetric polynomial in the roots. In particular:

\(r_{1} + r_{2} + \hdots + r_{n} = \frac{- a_{n - 1}}{a_{n}}\) \(r_{1} r_{2} \hdots r_{n} = \frac{( - 1 )^{n} a_{0}}{a_{n}}\)

The quadratic case is treated in detail in the page on trinomials.

Finding roots: an overview of methods

For polynomials of degree 1 and 2, exact formulas are elementary. A linear polynomial $a x + b$ has the unique root $x = - b / a$. For a quadratic $a x^{2} + b x + c$, the roots are given by the quadratic formula:

The quantity $\Delta = b^{2} - 4 a c$ is the discriminant.

• If $\Delta > 0$ the polynomial has two distinct real roots.
• If $\Delta = 0$ it has one real root of multiplicity 2.
• If $\Delta < 0$ it has two complex conjugate roots.

Closed-form solutions also exist for degree 3 (Cardano’s formula) and degree 4 (Ferrari’s method), though they are considerably more involved. For higher degrees, the problem requires more advanced techniques.

Roots of a polynomial are precisely the solutions to the corresponding polynomial equation $p ( x ) = 0$, and the methods outlined above apply directly to both settings.

An important application of polynomial roots occurs in partial fraction decomposition, where a rational function $P ( x ) / Q ( x )$ is expressed as a sum of simpler terms. The structure of these terms is determined by the roots and multiplicities of the denominator $Q ( x )$. Simple roots of $Q ( x )$ correspond to distinct linear factors, whereas repeated roots result in sequences of terms with increasing order.

Selected references

• University of Maryland,m C. D. Levermore. Formulas for Roots of Polynomials
• Stanford University, S. Boyd. Rational Functions and Partial Fraction Expansion
• Northeastern University. Polynomial Functions, Factorization in $F [ x ]$