Coefficient of xy(z^-2) in (x-2y+3(z^-1))^4
The Attempt at a Solution
I was wondering if anyone could give me an explanation for my answer?
The coeffecient of xy(z^-2) does not = 4 where I would be able to use the multinomial theorem.
So since I see that the z in xy(z^-2) and the z in (x-2y+3(z^-1))^4 are both negative, I can treat them as positives
a = x
b = -2y
c = 3z
so I have (a+b+c)^4, 4!/2! = 24abc^2 = 24(x)(-2y)(3z)^2
but remember that z is negative so the answer is -216xyz^-2
This is how I did it, but I would just like an explanation on why I can consider z positive.
So if anyone can answer these two questions:
It seems like if I did the expansion by hand z will always be negative anyways, so I can just treat z as a positive and find the coefficient like I would any other problem and just change it to a negative exponent at the end. Does my reasoning seem alright here?
Also, there would be no answer to this problem if it was xy(z^2) instead of xy(z^-2), correct?