# Roots of a Polynomial
Source: algebrica.org — CC BY-NC 4.0
https://algebrica.org/roots-of-a-polynomial/
## Definition
Let \\(p(x)\\) be a [polynomial](../polynomials.md) 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](../sign-analysis-in-inequalities.md) 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](../maximum-minimum-and-inflection-points.md) appearance.
- - -
These properties follow from the local factorization:
\\[
p(x) = (x - r)^m q(x)
\\]
with \\(q\(r\) \neq 0\\). Since \\(q\\) is [continuous](../continuous-functions.md) 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](../derivatives.md) 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](../polynomial-equations.md) 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](../synthetic-division.md).
- - -
## The fundamental theorem of algebra
In the field of [complex numbers](../complex-numbers-introduction.md) \\(\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](../fields.md). 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](../trinomials.md).
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## 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](../polynomial-equations.md) \\(p(x) = 0\\), and the methods outlined above apply directly to both settings.
An important application of polynomial roots occurs in [partial fraction decomposition](../partial-fraction-decomposition.md), 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.
