Question

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

• A. $7,-2$
• B. $-7,2$
• C. $7,2$
• D. $-7,-2$

Answer 1

Answer: (C)

$7,2$

$x=-9$ is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
applying $R_1\to R_1+R_2+R_3$ and expanding gives
$\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=(x+9)\left|\begin{array}{ccc}1& 1& 1\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
$\Rightarrow (x+9)(x^2-9x+14)=0$
$\therefore 7,2$ are the other roots.