Question

f(x) is a polynomial such that f(1) = 8 and f(3) = 16. r(x) is the remainder when f(x) is divided by (x-1)(x-3). What is r(5)?

Answer 1

we know that r(x) = ax+b, since the divisor is of degree 2.

So, f(x) = h(x)(x-1)(x-3) + ax+b
f(1) = a+b = 8
f(3) = 3a+b = 16
now it's easy:
2a = 8, so a=4,b=4

r(x) = 4x+4, so r(5) = 24