Polynomials!!!

Since the $G.C.D.$ of the polynomials is $x - 2$ , their remainders should be $0$ .

Given that $x - 2$ is the $G.C.D.$ , equating it to $0$ , $x = 2$

Substitute $x = 2$ , first into $x^3$ - $3$ $x^2$ + $px$ + $24$ = $0$ , where $p = -10$ ,

then into $x^2$ - $7x$ + $q$ = $0$ , where $q$ = $10$ .

Since we are looking for the value of $p$ + $q$ , - $10$ + $10$ = $\boxed{0}$ , which is the answer .

As Mr Alvaro has already mentioned, that $x-2$ is the GCD of the two other polynomials implies that when each is divided by it the remainders must be zero.

Using the Python sympy library:

code seen in interactive mode

The remainder for the first polynomial is $2p+20$ , that for the second $q-10$ , implying values of -10 and 10 for $p$ and $q$ respectively.

Use the sum of the roots to find the second root of the quadratic equation. Multiply the roots to get q. Divide the cubic equation by x-2 and substitute the values of p that would leave a remainder of 0. It turns out p=-10 and q=10.