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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.

01-04-2007 01:16:19
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