Problem:
Let P(x) be a polynomial such that when P(x) is divided by x−19, the remainder is 99, and when P(x) is divided by x−99, the remainder is 19. What is the remainder when P(x) is divided by (x−19)(x−99)?
Answer Choices:
A. −x+80
B. x+80
C. −x+118
D. x+118
E. 0
Solution:
From the hypothesis, P(19)=99 and P(99)=19. Let
P(x)=(x−19)(x−99)Q(x)+ax+b
where a and b are constants and Q(x) is a polynomial. Then
99=P(19)=19a+b and 19=P(99)=99a+b
It follows that 99a−19a=19−99, hence a=−1 and b=99+19=118. Thus the remainder is −x+118.