Roots of a Polynomial

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Source: algebrica.org — CC BY-NC 4.0 https://algebrica.org/roots-of-a-polynomial/

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:

p(r) = 0

Given a polynomial of the form:

p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

with $a_n \neq 0$, the element $r$ is a root precisely when the substitution $x = r$ produces the value $p(r) = a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0$. The terms root and zero are used interchangeably.

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

For a root of 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 of odd multiplicity greater than one, that is $m \geq 3$, the graph crosses the axis but appears flatter at the intercept. The flattening becomes more pronounced as the multiplicity increases, giving the curve an inflexion-like appearance.

These properties follow from the local factorization:

p(x) = (x - r)^m q(x)

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

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 $p \equiv q$.

Multiplicity of a root

The notion of multiplicity refines the definition of a root by quantifying how many times a given value is a root. Let $p(x)$ be a polynomial with coefficients in a field $\mathbb{F}$, and let $r \in \mathbb{F}$ be a root of $p(x)$. The multiplicity of $r$ is the largest positive integer $m$ such that $(x - r)^m$ divides $p(x)$ in $\mathbb{F}[x]$, while $(x - r)^{m+1}$ does not. Equivalently, $p(x)$ admits the factorization:

p(x) = (x - r)^m q(x)

with $q(r) \neq 0$. The polynomial $q(x)$ collects all the remaining factors of $p(x)$, and the condition $q(r) \neq 0$ guarantees that the exponent $m$ cannot be increased.

A root of multiplicity one is called a simple root. A root of multiplicity two or more is called a multiple root, with specific names attached to the lowest cases: a root of multiplicity two is a double root, a root of multiplicity three is a triple root. The sum of the multiplicities of all the roots of a polynomial of degree $n$ cannot exceed $n$. When equality holds, the polynomial decomposes completely into linear factors over $\mathbb{F}$:

p(x) = a_n (x - r_1)^{m_1} (x - r_2)^{m_2} \cdots (x - r_k)^{m_k}

with $m_1 + m_2 + \cdots + m_k = n$. Over the field of complex numbers, the fundamental theorem of algebra ensures that this complete decomposition always exists.

The multiplicity admits a differential characterization in terms of the derivatives of $p(x)$. The element $r$ is a root of multiplicity $m$ of $p(x)$ if and only if:

p(r) = p'(r) = p''(r) = \cdots = p^{(m-1)}(r) = 0

and

p^{(m)}(r) \neq 0

This criterion provides a constructive method for determining the multiplicity of a known root: successive derivatives of $p(x)$ are evaluated at $r$ until the first nonzero value is obtained, and the order of that derivative coincides with the multiplicity.

The differential characterization explains the graphical behaviour described above. At a simple root, the polynomial vanishes but its derivative does not, so the graph crosses the $x$-axis with nonzero slope. At a root of multiplicity $m \geq 2$, the first $m-1$ derivatives also vanish at $r$, and the graph becomes increasingly flat at the intercept as $m$ grows.

Rational root theorem

Given a polynomial with integer coefficients:

p(x) = a_n x^n + \cdots + a_0 \in \mathbb{Z}[x]

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 $q \mid a_n$.

The theorem 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 $\mathbb{C}$:

p(x) = a_n (x - r_1)^{m_1}(x - r_2)^{m_2} \cdots (x - r_k)^{m_k}

where $m_1 + m_2 + \cdots + m_k = n$. Counting roots with their multiplicities, a degree-$n$ polynomial has exactly $n$ roots in $\mathbb{C}$. This property characterizes $\mathbb{C}$ 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 $\mathbb{R}$:

(x - r)(x - \bar{r}) = x^2 - 2\alpha x + (\alpha^2 + \beta^2)

Every real polynomial of odd degree therefore has at least one real root. The factored form also establishes a direct relationship between roots and coefficients. Expanding the product:

a_n(x - r_1)(x - r_2)\cdots(x - r_n)

and comparing with the standard form:

a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0

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

r_1 + r_2 + \cdots + r_n = \frac{-a_{n-1}}{a_n}
r_1 r_2 \cdots 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 $ax + b$ has the unique root $x = -b/a$. For a quadratic $ax^2 + bx + c$, the roots are given by the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The quantity $\Delta = b^2 - 4ac$ 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.

The 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.