rational root theorem

English

Proper noun

rational root theorem

1. (algebra) A theorem which states a constraint on rational solutions of a polynomial equation with integer coefficients.

   Synonyms: rational root test, rational zero theorem, rational zero 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 $\mathbb {Q} [x]$.

   • 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

• French: please add this translation if you can
• German: Satz über rationale Nullstellen m
• Italian: teorema delle radici razionali 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