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 .

### Roots Calculator

 The Remainder Theorem How to do Long Division with Remainders Taylor Polynomial Remainder Bay's Theorem Greenes Theorem Parallelogram Theorems
 Calculator for Pythagorean Theorem Calculate Compound Interest Calculator Calculate Simple Interest Calculator Calculate Standard Deviation Calculator A Division Calculator A Fraction Calculator
 remainder theorem calculator polynomial solver perfect square trinomial definition dividing polynomials calculator taylor polynomial calculator