Rational root theorem

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on <a href="/facts/Rational_number/VJQmyY05">rational</a> <a href="/facts/Equation_solving/2RKaXrDM">solutions</a> of a <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equation</a>

a
          
            n
          
        
        
          x
          
            n
          
        
        +
        
          a
          
            n
            −
            1
          
        
        
          x
          
            n
            −
            1
          
        
        +
        ⋯
        +
        
          a
          
            0
          
        
        =
        0
      
    
    {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}

with <a href="/facts/Integer/PUwwrawx">integer</a> coefficients 
 
 
 
 
 a
 
 i
 
 
 ∈
 
 Z
 
 
 
 {\displaystyle a_{i}\in \mathbb {Z} }
 
 and 
 
 
 
 
 a
 
 0
 
 
 ,
 
 a
 
 n
 
 
 ≠
 0
 
 
 {\displaystyle a_{0},a_{n}\neq 0}
 
. Solutions of the equation are also called <a href="/facts/Root_of_a_polynomial/mlK3dhoT">roots</a> or zeros of the <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> on the left side.
The theorem states that each <a href="/facts/Rational_number/VJQmyY05">rational</a> solution ⁠
 
 
 
 x
 =
 
 
 
 p
 q
 
 
 
 
 
 {\displaystyle x={\tfrac {p}{q}}}
 
⁠ written in lowest terms (that is, p and q are <a href="/facts/Relatively_prime/bnCxCXxs">relatively prime</a>), satisfies:

<ul><li>p is an integer <a href="/facts/Divisor/DKNa2GvC">factor</a> of the <a href="/facts/Constant_term/KRFu0aum">constant term</a> a0, and</li>
<li>q is an integer factor of the leading <a href="/facts/Coefficient/5Hwq2Wuq">coefficient</a> an.</li></ul>
The rational root theorem is a special case (for a single linear factor) of <a href="/facts/Gauss%27s_lemma_(polynomial)/gATpSMek">Gauss's lemma</a> on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is an = 1.

Rational root theorem open-in-new

Rational root theorem