<s>[INST] Using factor theorem on a multivariable function Q: Use the factor theorem to show that $a + b - c$ is a factor of $f(a, b, c) = (a+b+c)^3 - 6(a + b + c)(a^2 + b^2 + c^2) + 8(a^3 + b^3 + c^3)$ and hence factorise completely.
I know that the factor theorem states for $f(x)$ if $f(a) = 0$ then $x-a$ is a factor but when there are multiple variables how do you use factor theorem?
 [/INST] Here is a compilation of my comments. Denote $c$ by $x$, write your polynomial as a cubic polynomial in $x$ whose coefficients are polynomials in $a,b$:
$$f(x)=f_0(a,b)x^3+f_1(a,b)x^2+f_2(a,b)x+f_3(a,b).$$
Plug in $x=a+b$, you get $f(a+b)=0$. So by the theorem, $x-(a+b)$ is a factor of $f(x)$: $f(x)=(x-(a+b))g(x)$. Then your original polynomial $f(c)=(c-a-b)g(c)=(a+b-c)(-g(c))$, so $a+b-c$ is a factor.
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