PlanetMath: rational root theorem

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rational root theorem

(Theorem)

Consider the polynomial

$\displaystyle p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$

where all the coefficients

If has a rational root where $\gcd(u,v)=1$ , then $u\vert a_0$ and $v\vert a_n$ .

This theorem is related to the result about monic polynomials whose coefficients belong to a unique factorization domain. Such theorem then states that any root in the fraction field is also in the base domain.

"rational root theorem" is owned by drini. [ full author list (3) | owner history (1) ]

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Keywords:

polynomial

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Cross-references: fraction field, unique factorization domain, monic polynomials, root, rational, integers, coefficients, polynomial
There are 4 references to this entry.

This is version 7 of rational root theorem, born on 2001-10-15, modified 2006-12-02.
Object id is 228, canonical name is RationalRootTheorem.
Accessed 19351 times total.

Classification:

AMS MSC:	12D05 (Field theory and polynomials :: Real and complex fields :: Polynomials: factorization)
	12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )

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