The process described in the previous section is somewhat tedious to divide two polynomials, so we use Remainder theorem.
Let be any degree polynomial greater than or equal to one and let \(a\) be any real number. When a linear polynomial divides , then will be the Remainder.
Let be any polynomial that is greater than or equal to degree \(1\).
Assume that the quotient is \(q(x)\) when is divided by and, the remainder is .
Applying the division algorithm, Dividend \(=\) Divisor \(\times\) Quotient \(+\) Remainder
......................... Equation (i).
Since the degree of is \(1\) and the degree of is less than the degree of , the degree of .
This means that is a constant, say .
So we can rewrite equation (i) as .................... Equation (ii)
In particular, if , then equation (ii) becomes .
Thus, \(p(a) = r\). Therefore, \(p(a)\) is the remainder.
Find the remainder when is divided by .
Solution: Zero of is \(1\), so as per remainder theorem remainder in this case will be \(p(x) =\) \(p(1)\) .
Consider the polynomial .
So that, the remainder of the polynomial is \(2\).