How is the nature of the roots of a third degree polynomial determined?

Question

Given a polynomial $p(x) = x^3-bx^2+cx-d = 0 $ such that all three roots are real positive integers. How does one figure out if the three roots are distinct? The coefficient of $x^3$ is 1. In the case of a quadratic equation, we can determine that the roots are distinct if $b^2-4ac != 0$ for the equation $ax^2+bx+c=0$. Are there similar methods that can be applied to a third-order polynomial?

Answer 1

One way to do it is to check if the discriminant of the cubic is positive. The roots are distinct $\iff$ the discriminant is nonzero.

For a cubic, our discriminant is $\Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd$. Yuck!

http://en.wikipedia.org/wiki/Discriminant

Answer 2

In this special case one may be able to avoid computing the discriminant. If you know all roots are natural numbers, the roots must be divisors of $d$. In fact, from $d=x_1x_2x_3$ and $c=x_1x_2+x_1x_3+x_2x_3$ we see that a multiple root must divide $\gcd(c,d)$. In simple cases one may simply try all these divisors.

Another approach is that multiple roots of $f$ are also roots of $f'$. So determine the polynomial $\gcd(f(x),f'(x))=\gcd(x^3-bx^2+cx-d,3x^2-2bx+c)=\ldots$.