Answer:
<em>D.) The remainder when p(x) is divided by x−3 is −2.</em>
Explanation:
If the polynomial p(x) is divided by a polynomial of the form x+k (which accounts for all of the possible answer choices in this question), the result can be written as
p(x)
/x+k = q(x) + r
/x+k
where q(x) is a polynomial and r is the remainder. Since x+k is a degree-1 polynomial (meaning it only includes x1 and no higher exponents), the remainder is a real number.
Therefore, p(x) can be rewritten as p(x)=(x+k)q(x)+r, where r is a real number.
The question states that p(3)=−2, so it must be true that
−2=p(3)=(3+k)q(3)+r
Now we can plug in all the possible answers. If the answer is A, B, or C, r will be 0, while if the answer is D, r will be −2.
A. −2=p(3)=(3+(−5))q(3)+0
−2=(3−5)q(3)
−2=(−2)q(3)
This could be true, but only if q(3) = 1
B. −2=p(3)=(3+(−2))q(3)+0
−2=(3−2)q(3)
−2=(−1)q(3)
This could be true, but only if q(3) = 2
C. −2=p(3)=(3+2)q(3)+0
−2=(5)q(3)
This could be true, but only if q(3) = −2
/5
D. −2=p(3)=(3+(−3))q(3)+(−2)
−2=(3−3)q(3)+(−2)
−2=(0)q(3)+(−2)
<em>
This will always be true no matter what q(3) is.</em>
<em>Of the answer choices, the only one that must be true about p(x) is D, that the remainder when p(x) is divided by x−3 is -2.</em>
