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Topic: Long polynomial (Read 533 times) |
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KicksGenius
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Posts: 31
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Re: Long polynomial
« Reply #1 on: Mar 15th, 2003, 6:17am » |
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There are 4 real roots to that 8th degree polynomial, I will not publish them at this time, because i only have the decimal approximations, and I assume that you would likeexact answers.
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NickH
Senior Riddler
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Re: Long polynomial
« Reply #2 on: Mar 15th, 2003, 6:33am » |
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Yes, exact answers only, please! Having said that, if you have fairly accurate answers, you can probably guess the exact roots, and thereby factorize the polynomial. (But I'm not saying that's the best approach...)
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Nick's Mathematical Puzzles
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KicksGenius
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Re: Long polynomial
« Reply #3 on: Mar 15th, 2003, 4:26pm » |
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Got them; 2-sqrt5, 2+sqrt5, 1-sqrt2, and 1+sqrt2 I believe those are the exact real roots, if not, well, My name isn't Kicks.
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SWF
Uberpuzzler
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Re: Long polynomial
« Reply #4 on: Mar 17th, 2003, 4:53pm » |
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Maybe there is an easier way, but for me, finding it would take longer than factoring: =(x4-6x3+6x2+6x+1)*(x4+2x3+ 2x2-2x+1) =(x2-2x-1)*(x2-4x-1)*( x2+(1-sqrt3)x+2-sqrt3)(x2+(1+sqrt3)x+2+sqrt3) The real roots are therefore 1+sqrt(2), 1-sqrt(2), 2-sqrt(5), 2+sqrt(5)
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wowbagger
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Re: Long polynomial
« Reply #5 on: Mar 18th, 2003, 6:08am » |
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Well, I don't know how easy factoring that polynomial is for you, but it isn't a piece of cake for me. After all, I'm not a computer algebra system. On the other hand, I haven't really tried yet. Somehow I have the feeling that there's a trick to it...
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NickH
Senior Riddler
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Posts: 341
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Re: Long polynomial
« Reply #6 on: Mar 18th, 2003, 1:58pm » |
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There is indeed a trick, of sorts. Here's a hint -- it involves making an appropriate substitution. (Substitutions are always "appropriate" in hindsight, aren't they?!)
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« Last Edit: Mar 18th, 2003, 1:59pm by NickH » |
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SWF
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Re: Long polynomial
« Reply #7 on: Mar 18th, 2003, 5:42pm » |
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I do not have access to computer algebra either, and waited to try this problem because I thought somebody would solve it that way. For me, factoring wasn't simple but was easier than figuring out the clever trick.
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