I have been trying to factor the polynomial
$$x^4 – 2x^3 + 5x^2 – 5x +1$$
but the only root I can find is $(x-1) $. The context is that this is the characteristic polynomial of a matrix whose Jordan Canonical Form I need. Any tips on getting the other complex roots?
Best Answer
HINT.
Factorise $(x-1)$ in prier to have a product $(x-1) P(x)$, where $P(x)$ is a third degree polynomial. Then you can use Cardano's formula to get the complex solutions.
For the records, when you have
$$P(x) = Ax^3 + Bx^2 + Cx + D$$
The solutions are provided by Cardano's method: first of all define the following quantities
$$p = B^2 - 3AC$$
$$q = 2B^3- 9 ABC + 27A^2D$$
$$\lambda = \sqrt[3]{\frac{q\pm \sqrt{q^2 - 4p^3}}{2}}$$
From this, solutions are:
$$x_1 = -\frac{1}{3A}\left(B + \lambda + \frac{p}{\lambda}\right)$$
$$x_2 = -\frac{1}{3A}\left(B - \frac{\lambda}{2}(1 - i\sqrt{3}) + \frac{p}{\lambda\left(-\frac{1}{2}(1 - i\sqrt{3})\right)}\right)$$
$$x_3 = -\frac{1}{3A}\left(B - \frac{\lambda}{2}(1 + i\sqrt{3}) + \frac{p}{\lambda\left(-\frac{1}{2}(1 + i\sqrt{3})\right)}\right)$$