linguistic:paraphrasing, stylistic:style_transfer
Proofs

All proofs below involve some mathematical analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This requirement has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.

Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root. This lemma is enough to establish the general case because, given a non-constant polynomial p with complex coefficients, the polynomial

has only real coefficients, and, if z is a root of q, then either z or its conjugate is a root of p. Here, p ¯ {\displaystyle {\overline {p}}} is the polynomial obtained by replacing each coefficient of p with its complex conjugate; the roots of p ¯ {\displaystyle {\overline {p}}} are exactly the complex conjugates of the roots of p

Many non-algebraic proofs of the theorem use the fact (sometimes called the "growth lemma") that a polynomial function p (z) of degree n whose dominant coefficient is 1 behaves like z when | z | is large enough. More precisely, there is some positive real number R such that

when | z | > R.