**Fundamental theorem of algebra.** Every non-constant polynomial with complex coefficients has at least one complex root.

We will not prove this theorem.

We can deduce that **every degree \(n\) polynomial has exactly \(n\) complex roots** by an inductive argument. A degree \(n\) polynomial has at least one root by the fundamental theorem of algebra, and by polynomial division, we can write it as a product of a linear factor and a degree \(n-1\) polynomial. By this degree \(n-1\) polynomial has at least one root, so this can be factorised to a linear factor and a degree \(n-2\) polynomial, and so on. By induction, a degree \(n\) polynomial has \(n\) complex roots.