Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number any polynomial is the sum of and the product by of a polynomial in of degree less than the degree of In particular, is the remainder of the Euclidean division of by and is a divisor of if and only if a property known as the factor theorem. Let . Polynomial division of by gives the quotient and the remainder . Therefore, . Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial by using algebraic manipulation: So, which is exactly the formula of Euclidean division. The generalization of this proof to any degree is given below in . The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that If the divisor is where r is a constant, then either R(x) = 0 or its degree is zero; in both cases, R(x) is a constant that is independent of x; that is Setting in this formula, we obtain: A constructive proofthat does not involve the existence theorem of Euclidean divisionuses the identity If denotes the factor of degree in this identity, and one has (since ). Adding to both sides of this equation, one gets simultaneously the polynomial remainder theorem and the existence part, in this case, of the theorem of Euclidean division. 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.