You know the quadratic formula, but what about the cubic formula: if

$\begin{displaymath}x^3 + px + q = 0\end{displaymath}$

then

$\begin{displaymath}x = \omega^i \sqrt[3]{ â \frac{q}{2} + \sqrt{\left( \frac{q......{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3}\right)^3}}\end{displaymath}$

(for

= 0, 1, or 2). Imagine having to memorize that for an exam!

There's also a quartic formula for fourth degree equations. You may have heard, however, that there is no formula to solve a quintic polynomial by adding, subtracting, multiplying, dividing, and taking roots of the coefficients. This was proved by the great Norwegian mathematican Niels Henrik Abel in 1824.

We'll talk about the elegant algebraic structures that encode information about solving polynomials, do a bit of basic group theory and Galois theory, and prove Abel's "impossibility theorem." Time permitting, we'll end with some intriguing mathematical puzzles.