Before you understand Remainder Theorem, you are advised to read:
What is Dividend, Divisor, Quotient & Remainder ?
What is Degree of a Polynomial ?
How to Find Value Polynomial ?
How to divide Polynomial by Binomial ?
(This Remainder Theorem applies to situation where a polynomial is divided by a linear polynomial)
Remainder Theorem says that, " If p(x) is divided by linear polynomial (x  a), then remainder will always be p(a) ".
Here, p(x) can be any polynomial of degree greater than or equal to zero. And a can be any real number.
Lets prove this Theorem with the help of following examples:
Example 1: Divide (x^{3} + 4) by (x  1)
Solution: As per the given question:
Polynomial = Dividend= (x^{3} + 4)
Linear Polynomial = Divisor = (x  1)
Division is done as shown below:
Hence, when (x^{3} + 4) is divided by (x  1), we get:
Quotient = (x^{2} + x + 1)
Remainder = 5
(Statement 1)
Now, let's apply remainder theorem on the same example:
Remainder Theorem says that, " If p(x) is divided by linear polynomial (x  a), then remainder will always be p(a) ".
Apply the values from given example and we get:
Polynomial p(x) = (x^{3} + 4)
Linear Polynomial (x  a) = (x  1)
So value of a = 1
And as per remainder theorem, x = a = 1
Now find the value of p(x) = (x^{3} + 4), where x = 1 and we get:
p(1) = (1)^{3} + 4
= 1 + 4 = 5
(Statement 2)
Hence, as per statement 1 and 2, Remainder is same i.e. 5.
This proves the remainder theorem.

