# 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).## Your turn...

Can you apply the Factor Theorem to decide whether the polynomial is divisible by the binomial ?

Can you apply the Factor Theorem to decide whether the polynomial is divisible by the binomial ?