Question

The determinant $\left|\begin{array}{ccc}4+x^2& -6& -2\\ -6& 9+x^2& 3\\ -2& 3& 1+x^2\end{array}\right|$ is not divisible by

• A. $x$
• B. $x^3$
• C. $14+x^2$
• D. $x^5$

Answer 1

Answer: (D)

$\left|\begin{array}{ccc}4+x^2& -6& -2\\ -6& 9+x^2& 3\\ -2& 3& 1+x^2\end{array}\right|$
$=(4+x^2)[(1+x^2)(9+x^2)-9]$
$+6[-6(1+x^2)+6]-2[-18+2(9+x^2)]$
$=(4+x^2)(9+9x^2+x^2{x}^4-9)$
$+6(-6-6x^2+6)-2(-18+18+2x^2)$
$=(4+x^2)(10x^2+x^4)-36x^2-4x^2$
$=40x^2+4x^4+10x^4+x^6-40x^2$
$=14x^4+x^6$
$=x^4(x^2+14)$
Which is not divisible by $x^5$ .