Square root of 2
Application of the rational root theorem

The irrationality of 2 {\displaystyle {\sqrt {2}}} also follows from the rational root theorem, which states that a rational root of a polynomial, if it exists, must be the quotient of a factor of the constant term and a factor of the leading coefficient. In the case of p (x) = x 2 − 2 {\displaystyle p(x)=x^{2}-2}, the only possible rational roots are ± 1 {\displaystyle \pm 1} and ± 2 {\displaystyle \pm 2}. As 2 {\displaystyle {\sqrt {2}}} is not equal to ± 1 {\displaystyle \pm 1} or ± 2 {\displaystyle \pm 2}, it follows that 2 {\displaystyle {\sqrt {2}}} is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when p (x) {\displaystyle p(x)} is a monic polynomial with integer coefficients; for such a polynomial, all roots are necessarily integers (which 2 {\displaystyle {\sqrt {2}}} is not, as 2 is not a perfect square) or irrational.

The rational root theorem (or integer root theorem) may be used to show that any square root of any natural number that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.
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