If the polynomial p(x) is divided by
x−3,
the result can be written as
=q(x)+.
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 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=2r
.
Therefore, the remainder when p(x) is divided by
x−3 is
−2Choice 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 negative, 2,need not be a root of p(x).