rational root theorem

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 ${\displaystyle p/q}$ is a root of the polynomial equation ${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}$, with ${\displaystyle a_{0},\ldots a_{n}\in \mathbb {Z} }$, then ${\displaystyle p\vert a_{0}}$ and ${\displaystyle q\vert a_{n}}$.

   • 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

     ${\displaystyle x^{3}+x+7}$

     is irreducible over ${\displaystyle \mathbb {Q} [x]}$.

   • 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.

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