Editing Hurwitz polynomial (section)

== Examples ==

A simple example of a Hurwitz polynomial is:

:<math>x^2 + 2x + 1.</math>

The only real solution is −1, because it factors as

:<math>(x+1)^2.</math>

In general, all [[quadratic polynomial]]s with positive coefficients are Hurwitz.
This follows directly from the [[quadratic formula]]:
:<math>x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.</math>
where, if the discriminant ''b''<sup>2</sup>−4''ac'' is less than zero, then the polynomial will have two [[complex conjugate|complex-conjugate]] solutions with real part −''b''/2''a'', which is negative for positive ''a'' and ''b''.
If the discriminant is equal to zero, there will be two coinciding real solutions at −''b''/2''a''. Finally, if the discriminant is greater than zero, there will be two real negative solutions,
because <math>\sqrt{b^2-4ac} < b</math> for positive ''a'', ''b'' and ''c''.