When the g(x) i.e. the divisor which cannot

When we divide a number by another number, we get a quotient and a
remainder. If the remainder is zero, then we say the numerator is divisible by
the denominator. Let us assume two numbers 17 and 3. We know that when we
divide 17 by 3, we get the quotient 5 and remainder 2. It can be expressed as
17= (3*5) +2. We can see that the remainder i.e. 2 is less than the divisor 3.
Similarly, when we divide 10 by 2, we get 10= (2*5) +0. So, the remainder is 0,
and we can say that 2 are a factor of 10 and 10 is a multiple of 2. The same
concept is used in polynomials. When a polynomial is divided by another
polynomial, we get a quotient which is in polynomial and a remainder. To make
some sums possible Remainder Theorem plays a vital role in finding factor or
multiple of those polynomials. The Remainder Theorem is a useful mathematical
theorem that can be used to factorize polynomials of any degree in a neat and
fast manner.

Definition

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The remainder theorem is an application of polynomial long
division. When we are dividing a polynomial with another polynomial it is being
expressed in the form: f(x)= g(x).q(x)+r(x), where f(x) is a polynomial, g(x)
is a divisor, q(x) is a quotient and r(x) would be the remainder Dividend =
(Divisor*Quotient) + Remainder.

Let us assume two polynomial p(x) and g(x) such that the degree
of polynomial p(x) would be greater than the g(x) i.e. the divisor which cannot
be zero. Then we can find the factor of polynomial by f(x) = g(x).q(x) + r(x)
where the remainder is lesser than the degree of divisor i.e. g(x). So, we can
say that f(x) divided by g(x) gives us the q(x) and r(x).

Remainder Theorem operates on the fact that a polynomial is
completely divisible by its factor to obtain a smaller polynomial then the
divisor and with the remainder having value smaller value or any real number.

Theorem

Let p(x) be any
polynomial of degree greater than or equal to one and is divided by the linear
polynomial x-a where a be any number which would be the divisor and we get the
value of x = a, then the remainder is p (a).

Proof

Let p(x) be any
polynomial of degree greater than or equal to 1. Suppose that when p(x) is
divided by x-a (where a is a divisor), the quotient is q(x) and the remainder
is r(x), i.e., p(x) = (x-a) q(x) + r(x) where Dividend = (Divisor x
quotient) + Remainder.

Since
the degree of x-a is less than the p(x) and the degree of r(x) is less than the
degree of x-a, where the degree of r(x) = 0. Which means that r(x) is constant,
say r. Therefore, p(x) = (x-a) q(x) +r.

By
putting the value in p(x) where x=a, this equation gives us p(a) = (a-a) q
(a) +r = r. Hence, this proves the theorem.