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That is why I went right to 2 quadratics in the numerators.
Hmmm, yes I see, but ought we not account for it. I tried taking the four outside, although perhaps that was not correct. What I am interested in, though, is why we don't have to worry about it when computing a,b,c,d,e and f
I think that is just some factor that Mathematica pulled out, for what reason... I do not know.
Oooh yes, that's very good - I'm still confused about that pesky four though
One way is to say the numerator is an 8th degree poly, a quadratic times by a six degree
Oooh thanks bobbym that's perfect, my only question would be where we've accounted for the fact that the product of our denominators is four times the original denominator. I also wonder if it would be possible to tell that our numerators would be quadratics if we hadn't had the answer to begin with - more out of curiosity than anything else.
Now what you do is form a 6 x 6 set of simultaneous linear equations. This is done by substituting x = -3,-2,-1,0,1,2 in the above. That wipes out the x and you are left with:
Which is exactly what we expected.
Oh of course - take your time - I hadn't meant for this thread to cause you - or anybody else - even more hard work.
Hmmm, I can definitely do 2) in theory - although that's not to say that I might not have some trouble with doing a large, difficult, practical example - although I should think I would probably enjoy trying and, well, as for 1) I not only know the form, I know what they are, since I wish only to verify it, as much as I would like to be able to derive something like this - not being a computer - or at least an extremely experienced and intelligent professor of advanced mathematics - I think that to do so may well be somewhere beyond me.
Oh, well, what I did was - mainly for my own satisfaction - to expand
Which of course gives a massive expansion, if it's of interest:
Which, when you add it all together gives:
Which, of course, is:
So I had satisfied myself with the denominators and I tried using my standard partial fractions method and said
But then I got:
And gave up, because I knew that I must have been on the wrong track. And, well, that's when I asked you lol