Question

If $\Delta =\left|\begin{array}{ccc}-x& a& b\\ b& -x& a\\ a& b& -x\end{array}\right|$ then, a factor of $\Delta$ is

• A. $a-b+x$
• B. $x^2-(a-b)x+a^2+b^2+ab$
• C. $x^2+(a+b)x+a^2+b^2-ab$
• D. $a+b+x$

Answer 1

Answer: (C)

$x^2+(a+b)x+a^2+b^2-ab$

$\Delta =\left|\begin{array}{ccc}-x& a& b\\ b& -x& a\\ a& b& -x\end{array}\right|$
$\Delta =-x(x^2-ab)-a(-bx-a^2)+b(b^2+ax)$
$\Delta =-x^3+3abx+a^3+b^3$
$\Delta =-x^3+3abx+(a+b{)}^3-3ab(a+b)$
$\Delta =(a+b{)}^3-x^3+3ab(x-a-b)$
$\Delta =(x-a-b)(x^2+(a+b\left)x\right)+a^2+b^2-ab$