Find the complex roots of this quartic polynomial

Question

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?

Answer 1

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:

$$x_1 = -\frac{1}{3A}\left(B + \lambda + \frac{p}{\lambda}\right)$$

$$x_2 = -\frac{1}{3A}\left(B - \frac{\lambda}{2}(1 - i\sqrt{3}) + \frac{p}{\lambda\left(-\frac{1}{2}(1 - i\sqrt{3})\right)}\right)$$

$$x_3 = -\frac{1}{3A}\left(B - \frac{\lambda}{2}(1 + i\sqrt{3}) + \frac{p}{\lambda\left(-\frac{1}{2}(1 + i\sqrt{3})\right)}\right)$$

Answer 2

By Descarte's rule of sign the change of sign in $f(x)=x^4-2x^3+5x^2-5x+1$ is $4$. Since you manually obtain one real root so it must have another positive real root. Check it. Thus two case arise: Two positive two complex or four positive. Oh.. Ethan Bolker Sir already mention the link of www.wolframalpha.com. Here you can find the roots numerically.

Answer 3

Since $x=1$ is a root of $x^4-2x^3+5x^2-5x+1$, then $x-1$ is a factor, and using polynomial long division gives the other factor to be $x^3-x^2+4x-1$.

Let the three remaining roots be $r_1$, $r_2$ and $r_3$. By Vieta's formulas, $r_1+r_2+r_3 = 2$, and $r_1 \cdot r_2 \cdot r_3 = 1$. By Saheb Pal's answer, $r_1$ is real, so $r_2$ and $r_3$ are conjugates of each other.

These are just some tips to check if your solution is correct. Since the roots are very messy, you can use numerical estimation (Newton's method), or Wolfram Alpha/a graphing calculator if you are allowed to.

Answer 4

you can factorise the original polynomial $(x - 1) (x^3 - x^2 + 4 x - 1)$ then try and solve the last factor. I would not actually bother with solving it by hand, and just use wolfram alpha or any other mathematical package. You can try and plot it first and then aproximate the solution with Newtons method. Because you know the polynomial, you can evaluate both the function value and it's derivative. Apparently you can also solve this by using an algebraic solution. We have $(x-1)(x + (\frac{1}{3}(1 - 11(\frac{2}{-7 + 3 \sqrt{597}})^{1/3} + (\frac{1}{2}(-7 + 3 \sqrt{597}))^{ 1/3})))(ax^2+bx+c)$ and then use abc formula