cubic

Cubic means degree three — a polynomial, equation, or function whose highest power is x^3. The word echoes a cube, whose volume is side^3, which is exactly where the third power earns its name. A general cubic looks like ax^3 + bx^2 + cx + d with a not equal to 0.

A cubic function's graph has a distinctive S-like sweep: it can rise, dip, and rise again (or the mirror image). Because the two ends of the graph head in opposite directions, a cubic always crosses the x-axis at least once, so every cubic equation has at least one real root.

Cubic equations can be solved exactly — there is a cubic formula, discovered in the 1500s — but it is cumbersome, so in practice you first hunt for an obvious root, factor out that (x - c), and reduce to a quadratic. This is one place synthetic division shines.