General fornula for roots of a $n-$th polynomial equation

Asked Nov 25 '22 at 11:44

Active Nov 25 '22 at 11:47

Viewed 44 times

Consider the following polynomial equation of degree $n$

$$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0x^0=0$$

I know one can solve for $n=2$ using the quadratic formula. And one can solve for $n=3$ directly using Cardano's formula, or throughout facotrizing into a quadratic equation. So my question is: Do we know a formula that can solve for roots of a $n-$th polynomial equation directly? What is it?

I searched too much time for this on google but I didn't find a formula and didn't find an answer denying its existence as well, it will be very helpful if someone tell me about it, Thank you guys.

edited Nov 25 '22 at 11:47

Sebastiano

7,649

asked Nov 25 '22 at 11:44

A.M.M Elsayed 马克

1

Look up Galois Theory. – coffeemath Nov 25 '22 at 11:49
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There is a very complicated formula by radicals (i.e a formula using the $4$ arithmetic operations and taking roots) for equations of degree $4$. Using Galois theory it can be proved that a formula by radicals simply doesn't exist for polynomials of degree $5$ or higher. – Mark Nov 25 '22 at 11:52
Some (random) posts on the topic Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?, Is a polynomial equation of degree $\ge 5$ not solvable by any way? – Sil Nov 25 '22 at 13:41

General fornula for roots of a $n-$th polynomial equation

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