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   Author  Topic: Long polynomial  (Read 533 times)
NickH
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Long polynomial  
« on: Mar 15th, 2003, 2:58am »
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Find all real roots of x8 - 4x7 - 4x6 + 4x5 + 38x4 - 4x3 - 4x2 + 4x + 1 = 0.
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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
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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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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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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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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.  Wink
 
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
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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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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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