The only difference between the forms in post #2 and #1 is that #2 is continuous at x=0.
That form has an additional real zero. Also, when you plug in numbers you will see the two forms are not equal.
]]>Apparently it is equivalent to...
For linear, quadratic, cubic and quartic polynomials, finding the roots is an easy task, because there are formulae for those roots in terms of the co-efficients of the polynomial and radicals. But this is in fact not true for polynomials of degree 5 or higher in general: the reasoning behind this is answered by Galois theory.
]]>The only difference between the forms in post #2 and #1 is that #2 is continuous at x=0.
Thank you for bringing integer relation algorithms to my attention!
It has honestly never occurred to me before that there might be numbers that can't be expressed in terms of known constants. It sounded counter-intuitive to me, until I realised that it does not imply that there are numbers which cannot be expressed in terms of defined operations. It merely means that it is sometimes necessary to expand our collection of "known constants".
]]>The form you produced in post #2 is not the same as post #1.
The two real roots are:
x = .046648023052674644930849814289785515159221216598276...
x = .31595555552492574948785882734255958583342547212420...
To get them obviously requires a CAS.
The procedure is to first establish a bound on the real roots using a Cauchy bound and or graphing, with a further refinement using Sturm sequences.. Then some iterative scheme has to be used to zero in on the roots once they have been isolated approximately.
To get closed forms requires the use of a PSLQ. But since most numbers can not be expressed in terms of known constants, this will often not be possible.
]]>It is not a practical problem since it is relatively easy to approximate the answers to a desired degree. I am just curious (and pedantic)
]]>There appears to be only two real roots. Finding analytical forms for them might be difficult or impossible.
]]>It is already simplified a little, but I am a bit lost now.
I am concerned more with answers than explanations at the moment. (:
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