A determinant containing a variable

Quark Itself

1. The problem statement, all variables and given/known data
I have this 4x4 determinant and usually these are just mechanical work, until I stumbled upon one containing x, how should one go about solving these type of determinants?
[x 2x 4 x ]
[1 2 2x 1 ]
[2x x-1 2 3x]
[ 2 x+1 x+3 x-1]
What are the different values of x?

2. Relevant equations



3. The attempt at a solution
I've tried ellimination and by using cofactors. When fully computed, it will be a polynomial of the 4th degree, as one probably can see.
Ellimination didn't make it simpler to factorize and cofactors just solved it until the polynomial popped up at the end-

Mark44

What did you get?

Quark Itself

8x⁴-22x³-22x²+47x+3

LCKurtz

Is this your determinant?$$
\left|\begin{array}{cccc}
x & 2x & 4 & x \\
1 & 2 & 2x & 1\\
2x & x-1 & 2 & 3x\\
2 & x+1 & x+3 & x-1
\end{array}\right|$$
If so, your last two terms should be ##44x+12##. Still, once you have the polynomial correct, you haven't said what you are supposed to do with it. It doesn't make sense to ask "what are the different values of ##x##".

HallsofIvy

The different values of x that give the determinant what property? That it be 0? That it not be 0?