General equation to find roots of a n-degree polynomial

[Sriman Dutta]

Is there any general equation to find the roots of an n-degree polynomial? Or is it not possible to construct such an equation?

[ajb]

It is impossible in general for polynomials of order 5 and above - see the Abel–Ruffini theorem which states there are no algebraic solutions of such polynomial equations.

 

Note that this does not mean there are no solutions (real or complex), just that you cannot write then in an algebraic form (in terms of radicals). Also it does not mean that you cannot find algebraic solutions to some polynomials of higher order.

 

You should also look up Galois theory which is the theory that deals with this question properly.

[studiot]

A good place to start looking is Ian Stewart's little book.

 

Galois Theory

 

Chapman and Hall

 

Professor Stewart has a well deserved reputation for making mathematics understandable, without loss of correctness.

 

You might also need

 

Introduction to Group Theory

 

W Ledermann

 

Longmans

 

since it is hard to do Galois without being doing Abel.

Edited August 17, 2016 by studiot

[John Cuthber]

...

 

You should also look up Galois theory which is the theory that deals with this question properly.

It's possible that he should "walk away" happy in the knowledge that the answer to his questions were "no" and "yes".

 

I can't speak for him but (as far as I'm concerned) knowing that we can only analytically solve polynomial equations up to quintics, is well within the sort of maths I understand: but Galois theory isn't.

 

Incidentally, it's sometimes possible to solve higher order polynomials by inspection.

Consider (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=0

[mathematic]

It's possible that he should "walk away" happy in the knowledge that the answer to his questions were "no" and "yes".

 

I can't speak for him but (as far as I'm concerned) knowing that we can only analytically solve polynomial equations up to quintics, is well within the sort of maths I understand: but Galois theory isn't.

 

Incidentally, it's sometimes possible to solve higher order polynomials by inspection.

Consider (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=0

You said up to quintics, it should have been up to quartics.

[John Cuthber]

You said up to quintics, it should have been up to quartics.

Oops!

[ajb]

It's possible that he should "walk away" happy in the knowledge that the answer to his questions were "no" and "yes".

You are right - so I we change my statement to 'one could look up Galois theory to understand the reasons why the answer is no'.

[Sriman Dutta]

Thanks......The Abel-Ruffini theorem is cool, nevertheless difficult to understand.

Does a similar limit exists for binomial and multinomial theorems?