what is exactly remainder theorm?

REMAINDER THEOREM

if P(x) is a polynomial in x then the remainder on dividing P(x) by x ? a is P(a)

PROOF:

Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a).

Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant.

Let r(x) = c, where c is some constant. Then

p(x) = (x–a) q(x) + c

Putting x = a in p(x) = (x–a) q(x) + c, we get

p(a) = (a–a) q(a) + c

? p(a) = 0 x q(a) + c

? p(a) = c

This shows that the remainder is p(a) when p(x) is divided by (x–a).