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.