# Cubic function

Missing image
Polynomialdeg3.png
Polynomial of degree 3

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

[itex]f(x)=ax^3+bx^2+cx+d,\,[itex]

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

[itex]y^2 = x(x-a)(x-b)\,[itex]

where [itex]0 < a < b[itex] 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

[itex]f(x) = \sqrt{x(x-a)(x-b)}\,[itex]

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

[itex](0,a) \cup (b,+\infty).\,[itex]

## 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

[itex]f(x) = ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3)\,[itex]

Let

[itex]q = \frac{{3c-b^2}}{{9}}[itex] and
[itex]r = \frac{{9bc - 27d - 2b^3}}{{54}}[itex]

Now, let

[itex]s = \sqrt{{r + \sqrt{{q^3+r^2}}}}[itex] and
[itex]t=\sqrt{{r-\sqrt{{q^3+r^2}}}}[itex]

The solutions are

[itex]x_1 = s+t-\frac{{b}}{{3}}[itex]
[itex]x_2=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}+\frac{{\sqrt{{3}}}}{{2}}(s-t)i[itex]
[itex]x_3=-\frac{{1}}{{2}}(s+t)-\frac{{b}}{{3}}-\frac{{\sqrt{{3}}}}{{2}}(s-t)i[itex]

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