Find the coefficient from the expansion (Explanation request

[Rijad Hadzic]

Homework Statement

Coefficient of xy(z^-2) in (x-2y+3(z^-1))^4

Homework Equations

The Attempt at a Solution

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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

= 24(-2)(9)xyz^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?

 

[kuruman]

c = 3z

You mean $c = 3z^{-1}$ of course. Your coefficient of -216 is correct. It would also be the coefficient of $xyz^2$ if you started from $(x-2y+3z)^4.$