In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division

Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...

. It states that the remainder

Remainder

In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....

 of a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

  divided by a linear

Linear

In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

 divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

  is equal to

Example

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

Proof

The polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor, quotient and remainder by, respectively, , , and , polynomial long division gives a solution of the equation
where the degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

 of is less than that of .

If we take as the divisor, giving the degree of as 0, i.e. :

Setting we obtain:

Applications

The polynomial remainder theorem may be used to evaluate by calculating the remainder, . Although polynomial long division

Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...

 is more difficult than evaluating the function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 itself, synthetic division

Synthetic division

Synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the formx - a,\ but the method generalizes to division by any monic polynomial...

 is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem

Factor theorem

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial f has a factor if and only if f=0....

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.