FANDOM


In algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial $ f(x) $ by a linear polynomial $ x-a $ is equal to $ f(a) . $ In particular, $ x-a $ is a divisor of $ f(x) $ if and only if $ f(a)=0. $[2]

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

$ f(x)=q(x)g(x) + r(x)\quad \text{and}\quad r(x) =0 \; \text{or}\; \deg(r)<\deg(g)\,. $

If we take $ g(x) = x-a $ 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

$ f(x)=q(x)(x-a) + r\,. $

Setting $ x=a $ in this formula, we obtain:

$ f(a)=r\,. $

A slightly different proof, which may appear to some people as more elementary, starts with an observation that $ f(x)-f(a) $ is a linear combination of the terms of the form $ x^k-a^k $, each of which is divisible by $ x-a $ since $ x^k-a^k=(x-a)(x^{k-1}+x^{k-2}a+\dots+xa^{k-2}+a^{k-1}) $.

The polynominal remainder theorem can be used as a shorter means to obtain a factor compared to polynomial long division.  

See also

Wikipedia.png 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.

References

  1. Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)". Formalized Mathematics 12 (1): 49–58. http://mizar.org/fm/2004-12/pdf12-1/uproots.pdf. 
  2. Larson, Ron (2014), College Algebra, Cengage Learning
Community content is available under CC-BY-SA unless otherwise noted.