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 by a linear polynomial is equal to In particular, is a divisor of if and only if ^{[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
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.
See also
- Synthetic division
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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