Question:
Given a polynomial p(x), the value of p(3) = 2.
Which of the following must be true for p(x)? (A) x  5 is a factor of p(x). (B) x  2 is a factor of p(x). (C) x + 2 is a factor of p(x). (D) The remainder when p(x) is divided by x  3 is 2. Answer:
If the polynomial p(x) is divided by x−3, the result can be written a
\[ \frac{p(x)}{x3} = q(x) + \frac{r}{x3}\]
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.
Therefore, p(x) can be rewritten as p(x) = (x−3)q(x) + r, where r is a real number. In the question it said that p(3) = −2 so it must be true that −2 = (3−3)q(3) + r =(0)q(3)+r =r or, r = 2 Therefore, the remainder when p(x) is divided by x−3 is −2. Answer is D.
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