Question

A certain polynomial P(x), x ∈ R when divided by x – a, x – b and x – c leaves remainders a, b and c respectively. Then find the remainder when P(x) is divided by (x – a) (x – b) (x – c) where, a, b, c are distinct.

Answer 1

By remainder theorem, P(a) = a, P(b) = b and P(c) = c

let the required remainder 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) – x must be zero polynomial (or identity)

Hence R(x) = x