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.
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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:
and
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.
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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.