Everyone who knows algebra is familiar with the quadratic formula for solving second degree equations.
It has been known since at least the 9th century.

The logical next step in algebra was to find the corresponding formula for the 3rd degree (cubic) equation. Starting about 1520, a group of Italian mathematicians competed to find the fabled cubic formula, and in 1545, one of them, Girolamo Cardano (1501 – 1576), published the solution. As we will see, the formula is complex, and of limited practical use. However, the cubic problem has great historical importance. Not only was the art of algebra was advanced by the effort to find it, but mathematicians eventually realized that the formula would only worked correctly if they faced up to the existence of complex numbers.

In this post, I will show a derivation of the formula, then show how it is used.

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Algebra classes always teach the Binomial Theorem – writing out the expanded form of a binomial raised to a power. What they don’t do is teach Newton’s brilliant generalization of the Binomial Theorem, usually called the Binomial Series, in which the exponent can be anything, not just an integer. The Binomial Series is much more important and interesting, and in this post I will describe it, including how Newton figured it out.

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