Definition:Quintic Equation
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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.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.27$: Abel ($1802$ – $1829$)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: quintic equation
