The fundamental theorem of algebra

The statement

d'Alembert–Gauss theorem. Every polynomial in $\mathbb{C}[X]$ of degree greater than or equal to $1$ has at least one root in $\mathbb{C}$.

It is also called the fundamental theorem of algebra. It expresses that the field $\mathbb{C}$ is algebraically closed: once we move from the reals to the complex numbers, we never again need to enlarge the set of numbers to solve a polynomial equation.

This is a sharp contrast with $\mathbb{R}$, where $X^2 + 1$ has no root, and with $\mathbb{Q}$, where $X^2 - 2$ has none either.

The factorization corollary

An induction on the degree turns the existence of one root into a complete factorization. If $P$ has degree $n \geq 1$ and leading coefficient $a_n$, then

$$P(X) = a_n \prod_{i=1}^{n} \left(X - z_i\right)$$

where $z_1, \ldots, z_n$ are the roots, repeated according to their multiplicity.

Proof. The theorem provides a root $z_1$. Euclidean division by $X - z_1$ gives $P = (X - z_1)\,Q + r$ with $r$ constant, and evaluating at $z_1$ shows $r = 0$. The quotient $Q$ has degree $n - 1$: we apply the induction hypothesis.

So remember: a polynomial of degree $n$ has exactly $n$ complex roots counted with multiplicity.

Polynomial	Degree	Complex roots with multiplicity
$X^2 + 1$	$2$	$i$, $-i$
$X^2 - 2X + 1$	$2$	$1$ double
$X^3 - 1$	$3$	$1$, $j$, $\overline{j}$
$X^5 - 3X^2 + 1$	$5$	five roots, not explicit

The last row is a reminder of what the theorem says and does not say: it asserts the existence of the roots, without providing any means of computing them. In fact, the Abel–Ruffini theorem shows that beyond degree $4$, no formula in radicals can exist in general.

Relations between coefficients and roots

The factorization gives Vieta's formulas for free. In degree $n$, by expanding the product:

$$\sum_{i=1}^{n} z_i = -\frac{a_{n-1}}{a_n} \qquad \text{and} \qquad \prod_{i=1}^{n} z_i = (-1)^n\,\frac{a_0}{a_n}$$

They let us work with the roots without ever computing them, which is often the only practical route.

Classic pitfall

Confusing "$n$ roots" with "$n$ distinct roots". The polynomial $X^2 - 2X + 1 = (X-1)^2$ has only one root, but it is a double root: the count of $n$ is only exact with multiplicities. A polynomial of degree $n$ has at most $n$ distinct roots, and exactly $n$ counted with multiplicity.