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 ## Proof

The polynomial remainder theorem follows from the definition 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 we take as the divisor, either r = 0 or its degree is zero; in both cases, r is a constant that is independent of x; 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 the terms of the form , each of which is divisible by since .

The polynominal remainder theorem can be used as a shorter means to obtain a factor compared to polynomial long division. This page uses content from Wikipedia. The original article was at Polynomial remainder theorem.The list of authors can be seen in the page history. As with the Math Wiki, the text of Wikipedia is available under the Creative Commons Licence.