2

I'm trying to solve a problem with multiple unknowns, and I've managed to put all of them in terms of $u$, where $u^8-u^4-u^2+u-35=0$. In fact, I only need the positive real zero of this polynomial, which is just above 1.59. However, WolframAlpha gives a slightly different numerical solution than I get from using Newton's method by hand (no doubt just an issue of precision) and in any case it would be nice to be able to get an exact solution in terms of radicals, or at least to find a simpler polynomial to solve. I know that exact solutions are difficult to find for polynomials over degree four, but is there anything I can do with this specific one?

Robert Glover
  • 21
  • 3
    This doesn't look like it factors (just plugged it into Maple), and there are no general formulas by radicals for polynomials of degree $\ge 5$, so I think you have to do it numerically. – Lukas Geyer Mar 01 '13 at 17:42
  • 1
    I'm getting 1.59, not 1.57. To be more precise 1.597765495 –  Mar 01 '13 at 17:46
  • 1
    maple gives 1.5977654945899746971 for positive root. A direct plugin gives left side at 1.597 is -1.4835, at 1.598 is +.0455 so I think the 1.57 is off; root is between 1.597 and 1.598. BTW maple12 could not solve in closed form at all. [guess Byron just said this... ] – coffeemath Mar 01 '13 at 17:52
  • Yes, the value of 1.59 etc. is correct, not 1.57. I copied it incorrectly when I typed up the question. Fixed above. – Robert Glover Mar 01 '13 at 20:22
  • 1.59776549458997469706225784132707983100658117340792837047528099395075826173931732873876858965706357577416956219434047186801478028278617314226206251454862394375267334944956072954558259485259243788505081160801427812613451221961395810703981480908364615137794261626671006239572279306374609455082288538397977104750050297905253980745455224227563279239729866081759236409992599138130781290804286118080657481869945698701186007686165605611646528799548369783088180458953013390750880716261310360713248617139534924766408836716048237920819672594729844711957540653988171400303985371153697124436607113092911409397 – dot dot Mar 01 '13 at 21:06
  • Using Galois theory, one finds that this equation can not be solved by radicals. – i. m. soloveichik Jul 12 '14 at 15:00

0 Answers0