Question

A certain polynomial `P(x)x in R` when divided by k`x-a ,x-ba n dx-c` leaves remainders`a , b ,a n dc` , resepectively. Then find remainder when `P(x)` is divided by `(x-a)(x-b)(x-c)w h e r eab, c` are distinct.

Answer 1

By remainder theorem, P(a) = a, and P(c) =c . Let the required reminder be R (x). Then,
P(x)= (x-a)(x-b)(x-c)Q(x)+R(x)
Where R(x) is a polynomial of degree at most 2.
We get R(a) = a, R(b) = b and R(c) = c.So, the equation R(x)- x = 0 has three roots a, b, and c, But its degree is at most 2. So, R(x) must be a zero polynomial (or identity). Hence, R(x) = x.