Set $F := F (X, Y, Z) = (X^2 + 3Y − Z^2)^8$. Determine the coefficients with which the following terms appear in $F$.
- $X^4 Y^2 Z^2.$
- $X^{10} Y^2 Z^2$.
I would know how to find the coefficient if it was just $(X^2+Y-Z^2)^8$. It is the coefficient of the $y$ term that is making me stuck. Any suggestions please?
The way I think about it is this. Instead of thinking about the coefficient of $Y$, think about the coefficient of $(3Y)$. Similarly think about the coefficient of $(-Z^2)$ instead of the coefficient of $Z$, and the coefficient of $(X^2)$ instead of the coefficient of $X$. So you write $$ (X^2 + 3Y - Z^2)^8 = ((X^2) + (3Y) + (-Z^2))^8 $$ Now for the first question, you need the coefficient of $$ X^4 Y^2 Z^2 $$ But what that really means is you need to find the term with $(X^2)^2 (3Y)^2 (-Z^2)^1$. But there is no such coefficient. Then for the next one, you need the coefficient of $$ X^{10} Y^2 Z^2 $$ But what this really means is you need to find the term with $(X^2)^5 (3Y)^2 (-Z^2)^1$. So first, find the coefficient of $a^5 b^2 c$ in $(a + b + c)^8$. Then just write $a = (X^2)$, $b = (3Y)$, and $c = (-Z^2)$, and find what the new coefficient is. Hint: the new coefficient will just be multiplied by some factors of 3 (from $b$) and -1 (from $c$).