A certain polynomial P(x) , x belongs to R , when divided by x-a , x-b and x-cleaves remainders a - Maths - Mathematical Reasoning

Question

A certain polynomial P(x) , x belongs to R , when divided by x-a ,
x-b and x-cleaves 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

Rajat Sethi · Accepted Answer

By remainder theorem,when P(x) is divided by (x-a),(x-b),(x-c) the remainders are P(a),P(b) and P(c) respectively When f(x) is divided by q(x) the remainder will be having degree less than q(x)∴ 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)Now, (i) - (ii) we havep(a^2 - b^2)+ q(a-b)=a-b ⇒p(a+b)+q=1-----(iv) (ii) - (iii) gives p(b+c)+q=1----(v) (iv)-(v) gives p(a-c)=0 ∴ p=0(because a,b,c are distinct), then q=1 and r=0 So remainder is x

A certain polynomial P(x) , x belongs to R , when divided by x-a , x-b and x-cleaves 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 .