Septic equation

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

where $a \neq 0$ .

A septic function is a function of the form

where $a \neq 0$ . In other words, it is a polynomial of degree seven. If $a = 0$ , then it is a sextic function ( $b \neq 0$ ), quintic function ( $b = 0, c \neq 0$ ), 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 or cubic function 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.

Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. To give an example of an irreducible but solvable septic, one can generalize the solvable de Moivre quintic to get,

...
Wikipedia