I tried solving $4z^4-8z^3+z^2-2z+1=0$ by myself then failed and used Wolfram for finding roots. Now my question is: can the roots of a polynomial with degree of at least $4$ be found with some tools rather than curve sketching or using a calculator?
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I hate to break you the news, but this question is not gonna cut it. For starters, you should read this guide. We also expect askers to do their homework and search the site before asking (also basic googling in advance is a must). That would give you hits like this. Also, the question is kinda broad, in that entire books can be written (and have been written) about the attempts, and the theory it lead to. – Jyrki Lahtonen Jun 06 '18 at 06:16
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@Jyrki Lahtonen is it ok now? – PiGuy Jun 06 '18 at 06:24
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@Jean-ClaudeArbaut Then you might want to do your part in quality maintenance. I'm sick of established users who cannot bother to search the site at all. – Jyrki Lahtonen Jun 06 '18 at 06:24
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One of the first hits. – Jyrki Lahtonen Jun 06 '18 at 06:25
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1Possible duplicate of Is there a general formula for solving 4th degree equations (quartic)? – Jean-Claude Arbaut Jun 06 '18 at 06:26
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Grrr, silently changing the question after answers have been given is a very poor idea. – Jun 06 '18 at 06:26
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@YvesDaoust That edit improved the question immensely. Now it is a focused one as opposed to "too broad". Why did you try and answer an open-ended version anyway? – Jyrki Lahtonen Jun 06 '18 at 06:27
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@YvesDaoust the question is exactly the same I just added a context to it – PiGuy Jun 06 '18 at 06:29
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@Jean-ClaudeArbaut Three answers, all from users who should know better, came before I had a chance to search. My first priority was to attempt to guide the newbie asker. – Jyrki Lahtonen Jun 06 '18 at 06:29
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@JyrkiLahtonen: because that was an interesting question, calling for original answers. The new one is poorer. – Jun 06 '18 at 06:30
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@YvesDaoust The answers to the first version proved it a duplicate of umpteen earlier ones. Nothing new was offered. – Jyrki Lahtonen Jun 06 '18 at 06:31
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@user180165: had I known beforehand that sketching was prohibited, I wouldn't have wasted my time. Please respect the answerers. – Jun 06 '18 at 06:32
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1@JyrkiLahtonen: still in rudeness mode, I see. – Jun 06 '18 at 06:34
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@YvesDaoust I'm a 12 year old newbie from Nepal – PiGuy Jun 06 '18 at 06:34
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@user180165: I do forgive you, then, don't worry. But you should be more specific about the kind of tools you are thinking of. Finding roots with "bare hands" is a little difficult. – Jun 06 '18 at 06:34
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@YvesDaoust So I'm unknown of what shouldn't be done..this won't be repeated again – PiGuy Jun 06 '18 at 06:35
2 Answers
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For quartic polynomials, there are Ferrari's method and Descartes' method. Above degree $4$, there is no algebraic method, according to the Abel-Ruffini theorem.
José Carlos Santos
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You can sketch the function by finding the roots of the derivative and computing the locations of the extrema. From there, draw and estimate the values of the roots.
This is a recursive procedure.
But you can't escape evaluating the polynomial(s) at some points, which requires some... stuff.