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
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References
- ↑ Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)". Formalized Mathematics 12 (1): 49–58. http://mizar.org/fm/2004-12/pdf12-1/uproots.pdf.
- ↑ Larson, Ron (2014), College Algebra, Cengage Learning