# Definition:Quintic Equation

## Definition

Let $\map f x = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$ be a polynomial function over a field $\mathbb k$ of degree $5$.

Then the equation $\map f x = 0$ is the general quintic equation over $\mathbb k$.

## Examples

### Example: $z^5 - 2 z^4 - z^3 + 6 z - 4 = 0$

The quintic equation:

$z^5 - 2 z^4 - z^3 + 6 z - 4 = 0$

has solutions:

$1, 1, 2, -1 \pm i$

## Also see

• Abel-Ruffini Theorem, which proves that, in general, a quintic equation can not be solved analytically.