4.3. Solving Special Polynomial Equations 127
Theorem of Algebra that every complex polynomial had a zero by actually
expressing a zero in terms of the coefficients. However, already in 1797,
Gauss had established this result by other means.
We will review some of the methods for solving polynomial equations,
and then turn to the Fundamental Theorem. While a proper formulation
of the proof of this theorem requires quite advanced mathematics, it is
possible to discuss the strategy behind the proof in such a way that the
reader, even without the necessary background in topology, is nevertheless
convinced of its plausibility.
Exercises
- In Exercise 2.2.4, it was shown that the number of zeros of a noncon-
stant polynomial over an integral domain (and, in particular, over
a field) cannot exceed its degree. This is true, even if each zero is
counted as often as its multiplicity. In general, a polynomial need not
have as many zeros as its degree might indicate. Provide examples of
quadratic polynomials over the following fields which have no zeros
in the field: Q, R, Z2, Zs, Zs, Zr. - Consider the case of polynomials over the field of complex numbers.
Verify that the number of zeros of a polynomial counting multiplicity
is equal to the degree of the polynomial in the following cases:
(a) the degree does not exceed 4;
(b) the polynomial is a reciprocal polynomial of degree not greater
than 9;
(c) the polynomial is tn - c, for some positive integer n and complex
number c.- Occasionally, trigonometry can be used to find the roots of a high de-
gree polynomial equation. For example, one problem in the American
Mathematical Monthly asked for the roots of the equation
An(y) = 2awhere
fn(y) = yn - n~“--~ + [n(n - 3)/2!]~“-~ +...+ (-qn(n - r - 1)(7a - r - 2)... (n - 2r + 1)/r!]y”-2’ +... ,
in particular when a = 1.(a) Solve the equation in the cases a = 1 and n = 2,3,4,6,8.
(b) Verify that cos # + i sin 0 and cos 0 - i sin 0 are the roots of the
quadratic equation t2 -yt+l=O,wherey=2cos8.