Octic equation

From HandWiki
Graph of a polynomial of degree 8, with 8 real roots (crossings of the x axis) and with 7 critical points. In general, depending on the number and vertical location of the local maxima and minima, the number of real roots could be 8, 6, 4, 2, or 0. The number of complex roots equals 8 minus the number of real roots.

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 x4. Octics of the form

[math]\displaystyle{ ax^8 +cx^6+ex^4+gx^2+k=0 }[/math] are compositions of a quartic and x2.


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


  1. James Cockle proposed the names "sexic", "septic", "octic", "nonic", and "decic" in 1851. (Mechanics Magazine, Vol. LV, p. 171)
  2. 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).