Fundamental Theorem of Algebra

Every non-constant polynomial has a complex root — ℂ is algebraically closed.

The idea

Theorem (Fundamental Theorem of Algebra).

Every non-constant polynomial $p(z)$ with complex coefficients has a root in $\mathbb{C}$: a number $z_0$ with $p(z_0) = 0$.

Over the real numbers the corresponding statement is false: $x^{2} + 1$ has no real root, since a real square is never negative. The theorem says that adjoining $i$ repairs every such failure at once — no polynomial equation with complex coefficients ever demands a further enlargement of the number system. A system in which every non-constant polynomial has a root is called algebraically closed.

One root produces the rest. Given the root $z_0$, factor $p(z) = (z - z_0)\,q(z)$ with $q$ of degree $n - 1$, and apply the theorem to $q$. Repeating this, a polynomial of degree $n$ splits into $n$ linear factors. A root whose factor appears $k$ times is a root of multiplicity $k$, and is counted $k$ times.

Corollary.

A polynomial of degree $n \ge 1$ with complex coefficients has exactly $n$ roots in $\mathbb{C}$, counted with multiplicity.

When the coefficients of $p$ are real, the non-real roots come in pairs: if $a + bi$ is a root, so is its conjugate $a - bi$, its mirror image across the real axis.

Ways to work on it

• Walkthrough. Why ℂ is algebraically closed; complex conjugate roots; counting roots by degree.
• Practice. Count complex roots by reading off the degree.
• Hardest. Find all complex roots of a real quadratic with negative discriminant.

Not sure where to start? Take the ten-question placement test.