A certain polynomial P(x), x€R when divided by x-a, x-b, x-c leaves remainders a,bb,c respectively. then find the remainder when P(x) is divided by (x-a)(x-b)(x-c) where a,b,c are distinct.

Using Remiander theorem
f(a) is the remainder when f(x) is divided by x-a
when f(x) is divided by q(x) the remainder will be having degree less than q(x) so when P(x) is divided by (x-a),(x-b),(x-c) the remainders are P(a),P(b) and P(c) respectively
therefore P(x)=(x-a)(x-b)(x-c)g(x) + (px^2+qx+r) where (px^2+qx+r)is remainder g(x) is the quotient
P(a)=pa^2+qa+r=a --- (I)
P(b)=pb^2+qb+r=b----(II)
P(c)=pc^2+qc+r=c-----(III)
Closely observing u can q=1 p and r =0
So remiander will be x
and Secondly you could have also solved equations by using II-I and III-I.