A certain polynomial P(x),x E R WHEN DIVIDED BY (x-a),(x-b) and (x-c) leaves remainder a,b,c.then find the remainder when P(x) is divided by (x-a)(x-b)(x-c) where a,b,c are distinct

it is clear that f(a) is the remainder when f(x) is divided by x-a
and 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
P(a)=pa^2+qa+r=a --- (I)
P(b)=pb^2+qb+r=b----(II)
P(c)=pc^2+qc+r=c-----(III)
(I) - (II) gives p(a^2 - b^2)+ q(a-b)=a-b =>p(a+b)+q=1-----(4)
(II) - (III) gives p(b+c)+q=1----(5)
(4)-(5) gives p(a-c)=0
therefore p=0(because a,b,c are distinct)
then q=1 and r=0
so remainder is x
thanks and regards
M.MURALIKRISHNA
askIITIANS FACULTY

The polynomial P(x) - x has roots a,b, and c Hence P(x) - x = (x-a)(x-b)(x-c) Q(x) for some polynomial Q(x) or P(x) = Q(x) (x-a)(x-b)(x-c) + x It immediately follows that the remainder when P(x) is divided by (x-a)(x-b)(x-c) is x

Thanks and Regards,
M.MURALIKRISHNA
askIITians faculty