Is there a general formula for the solutions of a polynomial equation of the form $$Ax^n + Bx^{n-1} + C = 0,$$ where $A$, $B$, $C$, and $n$ are constants?
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1A general formula for the solutions of that polynomial? Is $N$ a real number or is it a natural number greater than $1$? – Workaholic Jan 11 '16 at 15:48
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Structurally, the equation you have is very similar to the one appear in Bring radical. At least when $N$ is a positive integer, you may be able to express $X$ as some sort of generalized hypergeometric function in $A,B,C$. – achille hui Jan 11 '16 at 16:14
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Yes, sorry N is positive integer – Andy Jan 11 '16 at 16:52
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The Galois group of the splitting field of, for example, $x^5 - x^4 + 1$, is isomorphic to $S_5$, which is not solvable, and hence the roots of that polynomial are not solvable in radicals. Hence, there is no general formula for the roots of a real polynomial of the form $A x^n + B x^{n - 1} + C$ in terms of radicals. (See the Wikipedia article on the Abel-Ruffini Theorem for more.)
Travis Willse
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