a polynomial f(x) when divided by x-1 and x-2 gives remainder 1 and 2 respectively. Then remainder when f(x) is divided by (x-1) and (x-2) is ?

Hello student,
In this question we shall be utilizing the fact that divident = divisor . quotient + remainder
We need to find out the remainder when f(x) is divided by (x-1)(x-2).
Let g(x) be the quotient and (ax+b) be the remainder in that case.
Then f(x) = (x-1)(x-2) g(x) + (ax+b)
It is given that when f(x) is divided by x-1 and x-2 it gives remainder 1 and 2 respectively.
So, f(1) = 2 and f(2) = 2
i.e. f(1) = a+b = 1
and f(2) = 2a+b = 2
Solving these two equations we have a = 1 and b = 0.
Hence the required remainder is ax+b = 1.