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 find its other two roots $a$ and $b$, and then find $ab$.

Answer 1

$\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
$\Rightarrow x(x^2-12)-3(2x-14)+7(12-7x)=0$
$\Rightarrow x^3-67x+126=0$
if $x=-9$
$(-9{)}^3-67(-9)+126=-729+729=0$
$\therefore (x+9)$ is the factor of $x^3-67x+126$
$\Rightarrow x^3-67x+126=(x+9)(x^2-9x+14)$
$\Rightarrow$ other two roots $a,b$ are $2,7$
$\therefore ab=14$