Sriman Dutta Posted August 17, 2016 Posted August 17, 2016 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 Posted August 17, 2016 Posted August 17, 2016 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 Posted August 17, 2016 Posted August 17, 2016 (edited) 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 Posted August 17, 2016 Posted August 17, 2016 On 8/17/2016 at 6:08 PM, ajb said: ... 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 Posted August 18, 2016 Posted August 18, 2016 On 8/17/2016 at 7:27 PM, John Cuthber said: 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 Posted August 18, 2016 Posted August 18, 2016 On 8/18/2016 at 12:05 AM, mathematic said: You said up to quintics, it should have been up to quartics. Oops!
ajb Posted August 18, 2016 Posted August 18, 2016 On 8/17/2016 at 7:27 PM, John Cuthber said: 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 Posted August 22, 2016 Author Posted August 22, 2016 Thanks......The Abel-Ruffini theorem is cool, nevertheless difficult to understand. Does a similar limit exists for binomial and multinomial theorems?
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