Problem: A polynomial p(x) has remainder three when divided by x−1 and remainder five when divided by x−3. The remainder when p(x) is divided by (x−1)(x−3) is
Answer Choices:
A. x−2
B. x+2
C. 2
D. 8
E. 15
Solution:
Let ax+b be the remainder when p(x) is divided by (x−1)(x−3), and let q(x),r(x) and t(x) be the quotients when p(x) is divided by (x−1). (x−3) and (x−1)(x−3), respectively. Then
p(x)=(x−1)q(x)+3p(x)=(x−3)r(x)+5p(x)=(x−1)(x−3)t(x)+ax+b
Substituting x=1 into the first and third equations and then substituting x=3 into the second and third equations yields
3=a+b5=3a+b.
Therefore, ax+b=x+2.