# Octic equation

In algebra, an **octic equation**^{[1]} 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.

## Solvable octics

Octics of the form

can be solved through factorisation or application of the quadratic formula in the variable *x*^{4}.

Octics of the form

can be solved using the quartic formula in the variable *x*^{2}.

## See also

## References

- ↑ James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (
*Mechanics Magazine*, Vol. LV, p. 171)