Fraction DecompositionI have the following problem:Suppose x + y + z = 0. Show that...
I have the following problem:
Suppose . Show that
I thought this was a fun problem to tackle, but I've been stuck on this for hours, and I can't seem to get anywhere. I've tried expanding, but no matter what I do, I just get a huge mess.
I also cannot find how to incorporate the initial assumption () anywhere. Have I missed something?
Answer & Explanation
You've surely been trying to express one of the unknowns using the others and plugging in. Try to preserve the symmetry between them instead:
But we know that the left hand side is a zero and, moreover, wherever something like appears we can replace that by a (again, strictly keeping exchange symmetry). So the same equation can be written as
and even better than that!
Surely replacing the right hand side by the left in your problem will make the multiplications simpler! Try to proceed from here :-)
Hint: apart from and , among which you have a first relation here, you'll need other symmetric forms like
Update: I have been able to derive both your equations this way so it's guaranteed to work. It just takes some time. There's a whole theory about monomial and power-sum symmetric polynomials but I didn't want to go into that.
Starting from , you can derive things like
and also some useful observations like
With these you're one little theorem away from and
The first identity does not involve much algebra, if you write
where the cancels out in the binomial exapnsion. A similar thing happends in the cubes term.
So the identity reads
and of the six terms on the right, two pairs add together, and you get the left side.