5. If the polynomials $x^{2}-3 x+2$ and $x^{2}-5 x+k$ have a common root, then the sum of all possible real values of $k$ is ( ).
(A) 3
(B) 4
(C) 5
(D) 6
(E) 10
5. E.

Since the polynomial
$$
x^{2}-3 x+2=(x-1)(x-2)
$$

has roots 1 and 2, therefore, $x^{2}-5 x+k$ has one root as $x=1$ or $x=2$.
When $x=1$, $k_{1}=4$;
When $x=2$, $k_{2}=6$.
Thus, $k_{1}+k_{2}=10$.