What is remainder theorem? With proof.

1.	What is remainder theorem? With proof.
Answer» Let p(x) be any polynomial whose degree is greater than or equal to one and a be any real number. If p(x) is divided by (x - a), then the remainder is equal to p(a).Proof:Let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x - a). Then,Let p(x) = (x - a) q(x) + r(x) ........1Where r(x)=0. But, (x - a) is polynomial of degree 1 and a polynomial of degree less than 1 is a constant. Therefore, either r(x) =0 or r(x) = constant.Let r(x) =r.then, P(x) =(x - a) q(x) + r. ......2Putting x=a in 2 we get P(a) =(a - a) q(a) +r=> p(a) = 0 * q(a)+r=> p(a) =0+r=> p(a) = rThis shows that the remainder is p(a) when p(x) is divided by (x - a).

Discussion