Polynomial remainder theorem


In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial is equal to In particular, is a divisor of if and only if a property known as the factor theorem.

Examples

Example 1

Let. Polynomial division of by gives the quotient and the remainder. Therefore,.

Example 2

Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial by using algebraic manipulation:
Multiplying both sides by gives
Since is the remainder, we have indeed shown that.

Proof

The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials and , asserts the existence of a quotient and a remainder such that
If the divisor is then either or its degree is zero; in both cases, is a constant that is independent of ; that is
Setting in this formula, we obtain:
A slightly different proof, which may appear to some people as more elementary, starts with an observation that is a linear combination of terms of the form each of which is divisible by since

Applications

The polynomial remainder theorem may be used to evaluate by calculating the remainder,. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.
The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.