In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. The remainder theorem is useful for evaluating a polynomial p(x) at value x = a .when a polynomial p(x) is divided by a factor x - a then we end up with a quotient polynomial q(x) and some remainder r.

Remainder theorem can be used to determine the zero of the polynomial . If dividing the polynomial by factor x - a gives the remainder r = 0 then x - a is one of the factor of the polynomial .In other words x = a is the zero of the polynomial.

Definition:

When you divide a polynomial f(x) by x - a the remainder r will be f(a) . Whenever we want to know the remainder after dividing by x-a  ,calculating f(a) will give the remainder.

## Remainder Theorem Proof

Suppose a polynomial f(x) is divided by the divisor polynomial g(x) to give quotient q(x) and remainder r(x) .

f(x) = q(x) $\times$ g(x) + r(x)

where degree of r(x) is one less than g(x) : the divisor.

If the divisor is x - a ,we take g(x) as x - a then r(x) becomes a constant r (since  its degree will be zero)
So f(x) = ( x - a ) $\times$ q(x) + r
For x = a  the equation becomes
f(a) = ( a - a ) $\times$ q(x) + r
f(a) = r
Hence proved .

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