GeoGebra Classroom

# Divisible Polynomials - Remainder and Factor Theorems

## Divisible Polynomials

A polynomial is divisible by a polynomial if there exists a polynomial such that , hence the remainder of the division of by is .

## Remainder Theorem

When the divisor polynomial is , the Remainder Theorem allows us to know a priori - that is without calculating the division - the remainder of the division of a polynomial by a binomial . This theorem states that the remainder is .

## Factor Theorem

The Factor Theorem uses the Remainder Theorem to provide us with a divisibility criterion for polynomials that can be very useful in applications: If , then the binomial is a factor of the polynomial . Conversely, if is a factor of , then

## Historical Notes

The Factor Theorem is sometimes referred to as the "Ruffini's Theorem", as well as the synthetic division algorithm for polynomials is sometimes named "Ruffini's Rule" because both are the results of the work of the Italian mathematician Paolo Ruffini (1765-1822).