Octic equation

In algebra, an octic equation is an equation of the form

where a ≠ 0.

An octic function is a function of the form

where a ≠ 0. In other words, it is a polynomial of degree eight. If a = 0, then it is a septic function (b ≠ 0), sextic function (b = 0, c ≠ 0), etc.

The equation may be obtained from the function by setting f(x) = 0.

The coefficients a, b, c, d, e, f, g, h, k may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.

Since an octic function is defined by a polynomial with an even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient a is positive, then the function increases to positive infinity at both sides; and thus the function has a global minimum. Likewise, if a is negative, the octic function decreases to negative infinity and has a global maximum. The derivative of an octic function is a septic function.