wu :: forums « wu :: forums - Long polynomial » Welcome, Guest. Please Login or Register. May 15th, 2024, 11:58pm

RIDDLES SITE WRITE MATH! Home Help Search Members Login Register

wu :: forums    riddles    easy (Moderators: SMQ, Grimbal, ThudnBlunder, Eigenray, Icarus, william wu, towr)    Long polynomial « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Long polynomial  (Read 524 times) NickH Senior Riddler     Gender: Posts: 341 Long polynomial   « on: Mar 15th, 2003, 2:58am » Quote Modify Find all real roots of x8 - 4x7 - 4x6 + 4x5 + 38x4 - 4x3 - 4x2 + 4x + 1 = 0. IP Logged Nick's Mathematical Puzzles KicksGenius Newbie     Gender: Posts: 31 Re: Long polynomial   « Reply #1 on: Mar 15th, 2003, 6:17am » Quote Modify 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. IP Logged NickH Senior Riddler     Gender: Posts: 341 Re: Long polynomial   « Reply #2 on: Mar 15th, 2003, 6:33am » Quote Modify 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...) IP Logged Nick's Mathematical Puzzles KicksGenius Newbie     Gender: Posts: 31 Re: Long polynomial   « Reply #3 on: Mar 15th, 2003, 4:26pm » Quote Modify 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. IP Logged SWF Uberpuzzler     Posts: 879 Re: Long polynomial   « Reply #4 on: Mar 17th, 2003, 4:53pm » Quote Modify 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) IP Logged wowbagger Uberpuzzler     Gender: Posts: 727 Re: Long polynomial   « Reply #5 on: Mar 18th, 2003, 6:08am » Quote Modify 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... IP Logged "You're a jerk, <your surname>!" NickH Senior Riddler     Gender: Posts: 341 Re: Long polynomial   « Reply #6 on: Mar 18th, 2003, 1:58pm » Quote Modify 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?!) « Last Edit: Mar 18th, 2003, 1:59pm by NickH » IP Logged Nick's Mathematical Puzzles SWF Uberpuzzler     Posts: 879 Re: Long polynomial   « Reply #7 on: Mar 18th, 2003, 5:42pm » Quote Modify 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. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------=> easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board