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Proof of the Fundamental Theorem of Algebra

Proof of the Fundamental Theorem of Algebra by way of
Multivariable Calculus
Jack Jacofsky
In mathematics, the fundamental theorem of algebra contends that each non-constant
single variable polynomial that has complex coefficients has more than one complex root (de
Oliveira 753). Stated algebraically, the theorem is expressed as follows:
F(z) = zn + a1 zn-1 + ... + an = 0;
where F is the complex polynomial.
In the above equation, f is a continuous function, The essence of the process is to prove
that if there is no complex root for f(z) = cnzn + cn-1zn-1+ · · · + c1z + c0, then f(z) is a (nonzero)
constant. Decompose f(z) into imaginary and real parts: f(z) = P(r, θ) + iQ(r, θ). From this
equation, P and Q are polynomials with constant terms that are independent of θ and have r
and n degrees. The constant terms are summarized as:
To prove the theorem, f must have no complex roots, meaning that P and Q are simultaneously
0 anywhere.
Jack Jacofsky
The angular component of the polar coordinates in f(z) = P + iQ is expressed as follows:
The derivative of the angular component is computed as follows:
The right sides of both derivative equations prove the fundamental theorem of algebra
because P 2 + Q 2≠ 0 for all the values of (r, θ). Therefore, this example proves that the angular
component (Q/P) is an expression of f=P + iQ. The proof of the Fundamental Theorem of
Algebra is anchored on the Gauss theorem of winding numbers.
Jack Jacofsky
Work Cited
de Oliveira, Oswaldo Rio Branco. "The Fundamental Theorem Of Algebra: From The Four
Basic Operations." The American Mathematical Monthly 119.9 (2012): 753.