cubic equation

A cubic equation is a polynomial equation of the third degree, the general form of which is


ax3 + bx2 + cx + d = 0,


where a, b, c, and d are constants.


There was a great controversy in 16th-century Italy between Girolamo Cardano and Niccoló Tartaglia about who should get credit for solving the cubic. At this time symbolic algebra hadn't been developed, so all the equations were written in words instead of symbols. Early studies of cubics helped legitimize negative numbers, give a deeper insight into equations in general, and stimulate work that eventually led to the discovery and acceptance of complex numbers. Cardano, in his Ars magna, found negative solutions to equations, but called them "fictitious". He also noted an important fact connecting solutions of a cubic equation to its coefficients, namely, that the sum of the solutions is the negation of b, the coefficient of the x2 term. At one other point, he mentions that the problem of dividing 10 into two parts so that their product is 40 would have to be 5 + v(–15) and 5 – v(–15). Cardano didn't go further than this observation of what later came to be called complex numbers, but a few years later Rafael Bombelli (1526–1672) gave several examples that involved these strange new mathematical beasts.