Fundamental theorem of algebra and its consequences

The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex root. In other words, if you have a polynomial like $P(x) = x^2 + 1$, there is always some complex number $z$ such that $P(z) = 0$. This means that complex numbers are 'enough' to solve any polynomial equation.

Important properties

• Every polynomial of degree $n \geq 1$ with complex coefficients has at least one complex root.

• A polynomial of degree $n$ can be factored into $n$ linear factors over the complex numbers.

• Some polynomials with real coefficients (like $x^2 + 1$) have no real roots, but always have complex roots.