Low Order Polynomials, Cubic Equations

Cubic Equations

When the polynomial has degree 3, divide by the lead coefficient and replace x with y-b/3a, as described in the previous section on quadratics. Now the cubic is monic, with no squared term. write the cubic as:

x³ + px + q = 0.

For a given solution x, let a and b be numbers satisfying a-b = x and ab = p/3. Combine these equations to give a quadratic, hence a and b can be derived, at least in theory, from x and p. Once a and b are found, the "other" ab pair, associated with x and p, is produced by swapping and negating a and b. Substitute x = a-b in the original polynomial to get,

a³ - b³ + q = 0.

Multiply through by 27a³, remembering that ab = p. The result is a quadratic in a³:

27a⁶ + 27qa³ - p³ = 0

Apply the quadratic formula, and take the cube root of the result. The 6 values of a group into three pairs, whose product is -3p and sum is x. All this can be expanded into a closed formula, but it would be messy. No matter, a computer can handle it.

This procedure won't work in rings with characteristic 2 or 3.