Editing Rational root theorem (section)

{{short description|Relationship between the rational roots of a polynomial and its extreme coefficients}}
In [[algebra]], the '''rational root theorem''' (or '''rational root test''', '''rational zero theorem''', '''rational zero test''' or '''{{math|''p''/''q''}} theorem''') states a constraint on [[rational number|rational]] [[Equation solving|solutions]] of a [[polynomial equation]]
<math display="block">a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0 = 0</math>
with [[integer]] coefficients <math>a_i\in\mathbb{Z}</math> and <math>a_0,a_n \neq 0</math>.  Solutions of the equation are also called [[root of a polynomial|roots]] or zeros of the [[polynomial]] on the left side.

The theorem states that each [[rational number|rational]] solution {{tmath|1=x=\tfrac pq}} written in lowest terms (that is,  {{math|''p''}} and {{math|''q''}} are [[relatively prime]]), satisfies:
* {{math|''p''}} is an integer [[divisor|factor]] of the [[constant term]] {{math|''a''<sub>0</sub>}}, and
* {{math|''q''}} is an integer factor of the leading [[coefficient]] {{math|''a<sub>n</sub>''}}.

The rational root theorem is a special case (for a single linear factor) of [[Gauss's lemma (polynomial)|Gauss's lemma]] on the factorization of polynomials.  The '''integral root theorem''' is the special case of the rational root theorem when the leading coefficient is&nbsp;{{math|1=''a<sub>n</sub>''&nbsp;=&nbsp;1}}.