rational root theorem

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English

English Wikipedia has an article on:

rational root theorem

Proper noun

rational root theorem

(algebra) A theorem which states a constraint on rational solutions of a polynomial equation with integer coefficients.
Synonyms: rational root test, rational zeros theorem, rational zeros test, p/q theorem

The rational root theorem states that if the rational number $p/q$ is a root of the polynomial equation $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0$ , with $a_{0},\ldots a_{n}\in \mathbb {Z}$ , then $p\vert a_{0}$ and $q\vert a_{n}$ .
- 2015, Marlow Anderson, Todd Feil, A First Course in Abstract Algebra, Taylor & Francis (CRC Press), 3rd Edition, page 50,
  Use the Rational Root Theorem 5.6 to argue that
  $x^{3}+x+7$
  
  is irreducible over $\mathbb {Q}$ .
- 2017, Martin H. Weissman, An Illustrated Theory of Numbers, American Mathematical Society, page 84:
  We demonstrate how to use the rational root theorem to prove the irrationality of many numbers.
- 2018, Mary Jane Sterling, Pre-Calculus For Dummies, Wiley, page 85:
  The rational root theorem has you create a list of possible rational roots using the lead coefficient and constant in the polynomial equation.

Translations

theorem in algebra

Catalan: teorema de les arrels racionals m
Finnish: rationaalijuuritesti
French: théorème des racines rationnelles m, théorème de la racine rationnelle m, théorème du zéro rationnel m
Galician: teorema das raíces racionais m, teorema dos ceros racionais m
German: Satz über rationale Nullstellen m
Italian: teorema delle radici razionali m
Portuguese: teorema do zero racional m, teorema das raízes racionais m
Spanish: teorema de los ceros racionales m, teorema de la raíz racional m, teorema de las raíces racionales m

Further reading

Factor theorem on Wikipedia.Wikipedia
Gauss's lemma (polynomials) on Wikipedia.Wikipedia
Descartes' rule of signs on Wikipedia.Wikipedia
Root-finding algorithms § Roots of polynomials on Wikipedia.Wikipedia
Real-root isolation on Wikipedia.Wikipedia
Geometrical properties of polynomial roots on Wikipedia.Wikipedia

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