Factoring Multivariable polynomials Determine a constant $k$ such that the polynomial
$$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$
is divisible by $x+y+z$.
Im having trouble utilizing the multivariable factor theorem for this problem>

Write these symmetric expressions in terms of $s = x+y+z$, $q = x y + x z + y z$, $p=x y z$. We get
\begin{eqnarray}
 x^5 + y^5 + z^5 &=& s^5 - 5 q s^3 + 5 q^2 s + 5 p s^2 - 5 p q\\
x^3 + y^3 + z^3 &=& s^3 - 3 q s + 3 p \\
x^2 + y^2 + z^2 &=& s^2 - 2 q
\end{eqnarray}
Therefore
$$x^5 + y^5 + z^5 + k (x^3 + y^3 + z^3)(x^2 + y^2 + z^2) \equiv - 5 pq + k (3p) \cdot (-2 q)= -(5 + 6k) pq \mod s$$
Hence $k = -\frac{5}{6}$. Check: 
$$x^5 + y^5 + z^5 -\frac{5}{6} (x^3 + y^3 + z^3)(x^2 + y^2 + z^2)
= \frac{1}{6}s^2 \cdot (s^3 - 5 q s + 15 p) $$
