Factor 9 terms with 3 variables into 4 expression

Question

I just got the determinant from a 4x4 matrix and the simplified version is below. $$ det(M) = \begin{vmatrix} 2k-mw^2 & -k & 0 & 0 \\ -k & 2k-mw^2 & -k & 0 \\ 0 & -k & 2k-mw^2 & -k \\ 0 & 0 & -k & 2k-mw^2\\ \end{vmatrix} $$ the polynomial I got after 1 hr is: $$ 5k^4 - 6k^2mw^2 + 6k^2m^2w^4 - 8k^3mw^2 + 2km^2w^4 - 4km^3w^6 - 3m^3w^6 + m^4w^8 - k^2m^2w^4 $$ I want to factor this out, I tried so many ways but just gave up.

Now I know, since it was defined in Physics that the system I am studying would produce 4 normal frequencies. Usually, the form appear as: $$ (k-mw^2)(3k-mw^2) \\ $$ (the example above is for 2x2 matrix, hence produces 2 normal frequencies)

which is very easy to solve for $ w $.

Please help, I just want to equate the whole equation to zero and get w.

Answer 1

The determinant of the matrix given is: $$(5 k^2-5 k m w^2+m^2 w^4) (k^2-3 k m w^2+m^2 w^4)$$

So, solving for $w$ in the equation $(5 k^2-5 k m w^2+m^2 w^4) (k^2-3 k m w^2+m^2 w^4)=0$ yields the following solutions (assume $m \ne 0)$:

$$w = \pm\sqrt{\frac{(3+\sqrt{5})}{2}} \sqrt{\frac{k}{m}}$$ $$w = \pm\sqrt{\frac{(5+\sqrt{5})}{2}} \sqrt{\frac{k}{m}}$$ $$w = \pm\sqrt{\frac{(3-\sqrt{5})}{2}} \sqrt{\frac{k}{m}}$$ $$w = \pm\sqrt{\frac{(5-\sqrt{5})}{2}} \sqrt{\frac{k}{m}}$$

Answer 2

First, let's make things a bit nicer to look at. Let's substitute $x = 2k - mw^2$:

$$ det(M) = \begin{vmatrix} x & -k & 0 & 0 \\ -k & x & -k & 0 \\ 0 & -k & x & -k \\ 0 & 0 & -k & x\\ \end{vmatrix} $$

I calculate the determinant to be $x^4 - 3k^2x^2 + k^4:$

$$ det(M) = x\begin{vmatrix} x & -k & 0 \\ -k & x & -k \\ 0 & -k & x\\ \end{vmatrix} + k\begin{vmatrix} -k & -k & 0 \\ 0 & x & -k \\ 0 & -k & x\\ \end{vmatrix} \\ = x^2\begin{vmatrix} x & -k \\ -k & x\\ \end{vmatrix} + xk\begin{vmatrix} -k & -k \\ 0 & x\\ \end{vmatrix} -k^2\begin{vmatrix} x & -k \\ -k &x\\ \end{vmatrix} \\ = x^4 - x^2k^2 - x^2k^2 - x^2k^2+ k^4 \\ = x^4 - 3k^2x^2 + k^4. $$

Using the quadratic formula on $x^2$ gives

$$x^2 = \frac{(3 \pm \sqrt{5})k^2}{2} = (2k - mw^2)^2.$$

Now take square roots of both sides, isolate $w$, and square root again, tossing the imaginary roots as you go along.