Editing Irreducible polynomial (section)

== Simple examples ==

The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

:<math>\begin{align}
  p_1(x) &= x^2 + 4x + 4\, = {(x + 2)^2}\\
  p_2(x) &= x^2 - 4\, = {(x - 2)(x + 2)}\\
  p_3(x) &= 9x^2 - 3\, = 3\left(3x^2 - 1\right)\, = 3\left(x\sqrt{3} - 1\right)\left(x\sqrt{3} + 1\right)\\
  p_4(x) &= x^2 - \frac{4}{9}\, = \left(x - \frac{2}{3}\right)\left(x + \frac{2}{3}\right)\\
  p_5(x) &= x^2 - 2\, = \left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right)\\
  p_6(x) &= x^2 + 1\, = {(x - i)(x + i)}
\end{align}</math>

Over the [[integer]]s, the first three polynomials are reducible (the third one is reducible because the factor 3 is not invertible in the integers); the last two are irreducible. (The fourth, of course, is not a polynomial over the integers.)

Over the [[rational number]]s, the first two and the fourth polynomials are reducible, but the other three polynomials are irreducible (as a polynomial over the rationals, 3 is a [[unit (ring theory)|unit]], and, therefore, does not count as a factor).

Over the [[real number]]s, the first five polynomials are reducible, but <math>p_6(x)</math> is irreducible.

Over the [[complex number]]s, all six polynomials are reducible.