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Cubic function

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

$f(x)=ax^3+bx^2+cx+d,\,$

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

$y^2 = x(x-a)(x-b)\,$

where $0 < a < b$ 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

$f(x) = \sqrt{x(x-a)(x-b)}\,$

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

$(0,a) \cup (b,+\infty).\,$

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 roots test or a numerical solution instead.

If we have

$f(x) = ax^3 + bx^2 + cx + d = (x - x_1)(x - x_2)(x - x_3)\,$

Let

$q = \frac{{3c-b^2}}{{9}}$ and

$r = \frac{{9bc - 27d - 2b^3}}{{54}}$

Now, let

$s = \sqrt[3]{{r + \sqrt{{q^3+r^2}}}}$ and

$t=\sqrt[3]{{r-\sqrt{{q^3+r^2}}}}$

The solutions are

$x_1 = s+t-\frac{{b}}{{3}}$

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

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

See also: cubic equation, spline.

Categories: Polynomials

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
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