A **sextic function** (sometimes called a *hexic function*) is a 6th degree polynomial function. In other words, it’s a polynomial where the highest degree (i.e. exponent) is 6.

The general form:

f(x) = x^{6} + a_{5}x^{5} + a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}

Where the coefficients a, b, c, d, e, f, g are usually integers, rational numbers, real numbers or complex numbers.

## Zeros of the Sextic Function

A sextic function can have between zero and 6 real roots/zeros (places where the function crosses the x-axis). These zeros can be difficult to find. In fact, roots of polynomials greater than 4 degrees (quartic equations) are notoriously hard to find analytically. Abel and Galois (as cited in Shebl) demonstrated that anything above a 4th degree polynomial can’t be solved with radicals; general sextics can be solved with Kampe de Feriet functions.

In theory, graphing the sextic function might be easier, but graphing can be equally challenging. This graph of the sextic function at Desmos.com has sliders so that you can experiment with different coefficients. You’ll notice though, that the graph is somewhat pathological, and honing in on the right window (so that you can see the entire graph) is a challenge in itself.

## Derivative of a Sextic Function

The first derivative of a sextic function is a quintic function.

## Sextic Equation

A sextic equation has almost the same notation as the general form of the sextic function, except that, instead of being presented with function notation, the formula is set equal to zero:

x^{6} + a_{5}x^{5} + a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} = 0

## References

Coble, A. (1911). The Reduction of the Sextic Equation to the Valentiner Form–Problem. Math. Ann. 70, 337-350.

Function graph: Krishnavedala [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]

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