Question

If x = -9 is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$, then other two roots are ___

Answer 1

Since $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
Expanding along $R_1$,
$x(x^2-12)-3(2x-14)+7(12-7x)=0\Rightarrow x^3-12x-6x+42+84-49x=0$
$\Rightarrow x^3-67x+126=0$ ....(i)
Here, $126\times 1=9\times 2\times 7$
For $x=2,2^3-67\times 2+126=134-134=0$
Hence, x = 2 is a root
For $x=7,7^3-67\times 7+126=469-469=0$
Hence, x = 7 is also a root.