In order to have three real roots, the function must have a local minimum and a local maximum. For that to happen, the derivative must be zero at those points.
The derivative of x^3 + 2px^2 + 2p^2 x + p is
3x^2 + 4px + 2p^2
To solve this for zero, the discriminant is 16p^2 - 24p^2, which is negative, indicating no real roots. Therefore the derivative is never zero, there are no local minumum nor maximum, and the cubic has only one real root.
Posted by Charlie
on 2008-03-04 11:13:44