For a polynomial p(x), the value of p(3) is −2. Which of the following must be...

Choice D is correct. If the polynomial p(x) is divided by x - 3, the result can be written as \(\frac{p(x)}{x - 3} = q(x) + \frac{r}{x - 3}\), where q(x) is a polynomial and r is the remainder. Since x - 3 is a degree 1 polynomial, the remainder is a real number. Hence, p(x) can be written as p(x) = (x - 3).q(x) + r, where r is a real number. It is given that p(3) = -2 so it must be true that -2 = p(3) = (3 - 3).q(3) + r = (0)q(3) + r = r. Therefore, the remainder p(x) is divided by x - 3 is -2.
Choice A is incorrect because p(3) = -2 does not imply that p(5) = 0. Choices B and C are incorrect because the remainder -2 or its opposite, 2, need not be a root of p(x).