Cubic function

In mathematics, a cubic function is a function of the form

<math>f(x)=ax^3+bx^2+cx+d,\,<math>

where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.

Bipartite cubics

The graph of

<math>y^2 = x(x-a)(x-b)\,<math>

where <math>0 < a < b<math> is called a bipartite cubic. This is from the theory of elliptic curves.

You can graph a bipartite cubic on a graphing device by graphing the function

<math>f(x) = \sqrt{x(x-a)(x-b)}\,<math>

corresponding to the upper half of the bipartite cubic. It is defined on

<math>(0,a) \cup (b,+\infty).\,<math>

Root-finding formula

The formula for finding the roots of a cubic function is fairly complicated. Therefore, it is common for some students to use the rational root test or a numerical solution instead.

If we have

<math>f(x) = ax^3 + bx^2 + cx + d = (x – x_1)(x – x_2)(x – x_3)\,<math>

Let

<math>q = \frac{{3c-b^2}}{{9}}<math> and

<math>r = \frac{{9bc – 27d – 2b^3}}{{54}}<math>

Now, let

<math>s = \sqrt[3]{{r + \sqrt{{q^3+r^2}}}}<math> and

<math>t=\sqrt[3]{{r-\sqrt{{q^3+r^2}}}}<math>

The solutions are

<math>x_1 = s+t-\frac{{b}}{{3}}<math>

<math>x_2=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}+\frac{{\sqrt{{3}}}}{{2}}(s-t)i<math>

<math>x_3=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}-\frac{{\sqrt{{3}}}}{{2}}(s-t)i<math>

See also: cubic equation, spline.