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P(x) upon division by (x-a)(x-b)(x-c) when P(a)=a, P(b)=b, P(c)=c

Source: 1990 ITAMO p3

January 31, 2020

polynomialremainderalgebra

Problem Statement

Let

a,b,c

be distinct real numbers and

P(x)

a polynomial with real coefficients. Suppose that the remainders of

P(x)

upon division by

(x-a), (x-b)

and

(x-c)

are

a,b

and

c

, respectively. Find the polynomial that is obtained as the remainder of

P(x)

upon division by

(x-a)(x-b)(x-c)