Question

Product of roots of equation $\left|\begin{array}{ccc}1+2x& 1& 1-x\\ 2-x& 2+x& 3+x\\ x& 1+x& 1-x^2\end{array}\right|=0$ is ____.

Answer 1

Answer: 0.5

$f\left(x\right)\left|\begin{array}{ccc}1+2x& 1& 1-x\\ 2-x& 2+x& 3+x\\ x& 1+x& 1-x^2\end{array}\right|=0$
Constant term $=f\left(0\right)=\left|\begin{array}{ccc}1& 1& 1\\ 2& 2& 3\\ 0& 1& 1\end{array}\right|$
$=2+2-2-3=-1$
Also coefficient of $x^4$ is '-2'
Hence product of roots is $p=1/2$