How do you prove the fundamental theorem of algebra?

We now prove the Fundamental Theorem of Algebra. g(z) = f(z + z0) f(z0) , for all z ∈ C. g(z) = bnzn + ··· + bkzk + 1, with n ≥ 1 and bk = 0, for some 1 ≤ k ≤ n. Let bk = |bk|eiθ, and consider z of the form z = r|bk|−1/kei(π−θ)/k, (2) with r > 0.

What is the fundamental theorem of algebra?

: a theorem in algebra: every equation which can be put in the form with zero on one side of the equal-sign and a polynomial of degree greater than or equal to one with real or complex coefficients on the other has at least one root which is a real or complex number.

When was the fundamental theorem of algebra proved?

1799
Fundamental theorem of algebra, Theorem of equations proved by Carl Friedrich Gauss in 1799.

What is the fundamental theorem of algebra example?

The fundamental theorem of algebra states the following: A polynomial function f(x) of degree n (where n > 0) has n complex solutions for the equation f(x) = 0. For example, the polynomial x^3 + 3x^2 – 6x – 8 has a degree of 3 because its largest exponent is 3. …

What are the fundamentals of algebra?

The fundamental theorem of algebra states that every non- constant single-variable polynomial with complex coefficients has at least one complex root . This includes polynomials with real coefficients, since every real number can be considered a complex number with its imaginary part equal to zero. Nov 4 2019

What is fundamental theorem?

Fundamental theorem. The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.

What is the theorem of algebra?

Definition of fundamental theorem of algebra.: a theorem in algebra: every equation which can be put in the form with zero on one side of the equal-sign and a polynomial of degree greater than or equal to one with real or complex coefficients on the other has at least one root which is a real or complex number.