Septic equation

For other uses, see Septic (disambiguation).
Graph of a polynomial of degree 7, with 7 real roots (crossings of the x axis) and 6 critical points. Depending on the number and vertical location of the minima and maxima, the septic could have 7, 5, 3, or 1 real root counted with their multiplicity; the number of complex non-real roots is 7 minus the number of real roots.

In algebra, a septic equation is an equation of the form

where a ≠ 0.

A septic function is a function of the form

where a ≠ 0. In other words, it is a polynomial of degree seven. If a = 0, then f is a sextic function (b ≠ 0), quintic 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 may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.

Because they have an odd degree, septic functions appear similar to quintic and cubic functions when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima). The derivative of a septic function is a sextic function.