Higher-Degree Polynomials - Irving, Ron (Beyond the Quadratic Formula, 2013)
We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four. Therefore, we will content ourselves with a survey of some central results about higher-degree polynomials, combining proof sketches (or no proofs at all) with historical discussions. The chapter ends with a proof of the fundamental theorem of algebra. Quintic Polynomials We have obtained the quadratic formula, Cardano's formula, and Euler's formula for solutions of polynomial equations up through degree four. What about quintic polynomials? By Theorem 1.14, quintics have real roots. It is natural to seek a formula for a root like our other formulas, one involving the polynomial's coefficients and the operations of sum, product, quotient, and nth root extraction. If we had such an expression, we could solve quintic equations by radicals, the word "radical" referring to an nth root. Through the seventeenth and eighteenth centuries, progress was made in simplifying the problem of solving quintic equations, but no solution by radicals was found. Mathematicians began to suspect that a solution may not exist, and this was shown to be true in the nineteenth century.