The fundamental theorem of algebra

The introduction of complex numbers definitively solves the problem of the solvability of polynomial equations.

Theorem — The fundamental theorem of algebra

Every non-constant polynomial with complex coefficients of degree $n$ admits exactly $n$ roots in $\mathbb{C}$, counted with multiplicity.

Consequence. Whereas in the reals the equation $x^2+1=0$ has no solutions, and in general a quadratic equation with $\Delta<0$ is impossible, in the complex numbers every quadratic polynomial equation has two solutions (possibly coincident). The solving formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ still holds, with the square root of a negative number understood in $\mathbb{C}$: $\sqrt{-d}=i\sqrt{d}$.

Example — A quadratic equation in $\mathbb{C}$

Solve $x^2+2x+5=0$ in $\mathbb{C}$. $\Delta = 4 - 20 = -16, \qquad \sqrt{\Delta} = 4i, \qquad x = \frac{-2\pm 4i}{2} = -1\pm 2i.$

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Topics: Complex numbers
Concepts: Complex number · Fundamental theorem of algebra
Skills: Solve equations · Use formulae