Octic equation

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

$ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+hx+k=0,\,$

where a ≠ 0.

An octic function is a function of the form

$f(x)=ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+hx+k,$

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

${\displaystyle ax^{8}+ex^{4}+k=0}$

can be solved through factorisation or application of the quadratic formula in the variable x⁴.

Octics of the form

${\displaystyle ax^{8}+cx^{6}+ex^{4}+gx^{2}+k=0}$

can be solved using the quartic formula in the variable x².

See also

• Cubic function
• Quartic function
• Quintic function

References

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