The Fundamental Theorem Of Algebra: Guaranteeing A Complex Solution To Every Polynomial Equation

the fundamental theorem of algebra

any polynomial with complex coefficients has at least one complex root

The fundamental theorem of algebra is a theorem in mathematics that states that every non-constant polynomial equation with complex coefficients has at least one complex root. In other words, every polynomial of degree one or higher has at least one complex root.

The theorem was first proved by the mathematician Carl Friedrich Gauss in 1799. It is one of the most important theorems in algebra and has many applications in various fields of mathematics and science.

The fundamental theorem of algebra can be stated formally as follows:

Let P(z) be a non-constant polynomial of degree n, where n is a positive integer, and let a1, a2, …, an be its complex coefficients. Then there exist complex numbers z1, z2, …, zn such that:

P(z) = a1(z – z1)(z – z2) …(z – zn)

That is, P(z) can be factored into a product of linear factors (z – zi) with complex roots zi. The number of distinct roots is exactly equal to the degree of the polynomial.

This theorem is important because it provides a guarantee that every polynomial equation with complex coefficients has at least one solution. It is also useful in finding the roots of polynomial equations and in solving problems in fields such as physics and engineering, where polynomial equations often arise.

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