Final answer:
By applying the Remainder Theorem to the given polynomials with the factor (x-2), we find that p = -10 and q = 10. Adding these together, we get p + q = 0. Therefore, the correct option is A. 0.
Step-by-step explanation:
If the G.C.D. (Greatest Common Divisor) of the polynomials x³−3x²+px+24 and x²−7x+q is (x−2), and we want to find the value of (p+q), we must apply polynomial division or use the Remainder Theorem. Because (x−2) is a factor of both polynomials, when we substitute x = 2 into the polynomials, they should both equal zero. This means:
- For the first polynomial: 2³−3(2)²+p(2)+24 = 0
- For the second polynomial: 2²−7(2)+q = 0
Calculating these, we get:
- 8∓12+2p+24 = 0 ⇒ 2p = −20 ⇒ p = −10
- 4−14+q = 0 ⇒ q = 10
Add p and q together to find p + q = −10 + 10 = 0.
Therefore, the answer is A. 0.