Hi /wsr/ I'm having trouble finding the partial fractions

>>372165
Where f(x)=fraction above

>>372165

With a polynomial lik this it's a long one but possible and standard.
x^8(a^2 - x^2) = -(x^8 * (x + a) * (x - a)), the minus can be writen before the fraction,
-(1 / x^8 * (x + a) * (x - a)), now use the standard way of creating the partial fractions for every power plus your two linar factors, meaning you will get ten fractions with coefficients, then make it into one fraction by main denominator, sort, find coefficients etc.

>>372184
I have calculated the numerators of 3 terms which are the terms coming from (a-x), (x+a) and the 8th term of x^8, that leaves another 7 numerators to be found which I can't figure out the way to do it

First observe that a^2 = (a^2 - x^2) + x^2.
Using that you get:

1 = a^(-2) (a^2 - x^2) + a^(-2) x^2
= a^(-2) (a^2 - x^2) + a^(-2) x^2 (a^(-2) (a^2 - x^2) + a^(-2) x^2)
= a^(-2) (a^2 - x^2) + a^(-4) x^2 (a^2 - x^2) + a^(-4) x^4
= a^(-2) (a^2 - x^2) + a^(-4) x^2 (a^2 - x^2) + a^(-4) x^4 (a^(-2) (a^2 - x^2) + a^(-2) x^2)
= a^(-2) (a^2 - x^2) + a^(-4) x^2 (a^2 - x^2) + a^(-6) x^4 (a^2 - x^2) + a^(-6) x^6
= a^(-2) (a^2 - x^2) + a^(-4) x^2 (a^2 - x^2) + a^(-6) x^4 (a^2 - x^2) + a^(-6) x^6 (a^(-2) (a^2 - x^2) + a^(-2) x^2)
= a^(-2) (a^2 - x^2) + a^(-4) x^2 (a^2 - x^2) + a^(-6) x^4 (a^2 - x^2) + a^(-8) x^6 (a^2 - x^2) + a^(-8) x^8

Now all that remains is to split the last term
2a = (a - x) + (a + x)
so
a^(-8) x^8 = (1/2) a^(-9) x^8 (a - x) + (1/2) a^(-9) x^8 (a + x)

>>372363
Thank you, I am impressed
Does a general rule to your method exist so I can study that too?

>>372165
Okay I'll take a whack in common place words and notions ( ideas )

1/ anything means you are trying to normalize a function, the bottom part, in order to include information that prioritizes the qualities therein. Therefore what we are trying to do is make sure we can use the info below and apply other functions to the part in particular that is at the top. The 1 there simple acting as an identity.

the function itself states that we have 8 degrees of information, that is the functions can and will vary over the 8 planes that the can adhere to. a^2 - x^2 means we are trying to differentiate between two seperate regions in a big cube thing that represents the 8 degrees. The issue with this is that the 8 degrees are not coplanar and so we will have to find a way to skew the idea of a cube to fit within the margins of the 8 degrees for x.

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kind of a thing and the parenthesis part will have to do with finding a mean for the square of the two degrees, meaning the part between those two values, to manage to squirm its way through the values represented by the 8 degrees.

Know that I avoid doing this kind of math because it's very dangerous information and people like me, poor people with little to lose, disappear after we help random people on the internet. I currently have several stalkers so I don't want to move forward or further because some of them are currently calling shots over my home ( someone just popped a gasket under my grass which they already killed ) and others are simply getting ready to interrupt me every time I start to think on how to explain it to you out of simple mathematical terms. So this concept I think should help you find a wedge between the different degrees.

I think I should also state that the median and mean have to ride a mean as central and as symmetric as possible so it'll be more like
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