# Octic equation

In algebra, an **octic equation**^{[1]} is an equation of the form

- [math]\displaystyle{ ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+hx+k=0,\, }[/math]

where *a* ≠ 0.

An **octic function** is a function of the form

- [math]\displaystyle{ f(x)=ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+hx+k, }[/math]

where *a* ≠ 0. In other words, it is a polynomial of degree eight. If *a* = 0, then *f* 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

By the Abel–Ruffini theorem, there is no general algebraic formula for a solution of an octic equation in terms of its parameters. However, some sub-classes of octics do have such formulas.

Trivially, octics of the form

- [math]\displaystyle{ x^8=a }[/math]

with positive *a* have the solutions

- [math]\displaystyle{ x_k=a^{1/8}\omega_k\, , \quad k=1, \dots , 8, }[/math]

where [math]\displaystyle{ \omega_k }[/math] is the *k*-th eighth root of 1 in the complex plane.

Octics which can be decomposed as a functional composition of solvable polynomials can be solved. For example, octics of the form

- [math]\displaystyle{ ax^8+ex^4+k=0 }[/math]

are compositions of a quadratic and *x*^{4}. Octics of the form

- [math]\displaystyle{ ax^8 +cx^6+ex^4+gx^2+k=0 }[/math] are compositions of a quartic and
*x*^{2}.

## Application

In some cases some of the quadrisections (partitions into four regions of equal area) of a triangle by perpendicular lines are solutions of an octic equation.^{[2]}

## See also

## References

- ↑ James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (
*Mechanics Magazine*, Vol. LV, p. 171) - ↑ http://forumgeom.fau.edu/FG2018volume18/FG201802.pdf Carl Eberhart, “Revisiting the quadrisection problem of Jacob Bernoulli”,
*Forum Geometricorum*18, 2018, pp. 7–16 (particularly pp. 14–15).