Question

If $a,b$ and $c$ are the roots of the equation $x^3+2x^2+1=0$, find $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

• A. $8$
• B. $-8$
• C. $0$
• D. $None$

Answer 1

Answer: (A)

$8$

$x^3+2x^2+1=0$

Now,

$a+b+c=-2$

$ab+bc+ca=0$

$abc=-1$

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=$

$=-\left(a^3+b^3+c^3-3abc\right)$

$=-\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)$

$=-\left(a+b+c\right)\left({\left(a+b+c\right)}^2-3ab-3bc-3ca\right)$

$=-\left(-2\right)\left[{\left(-2\right)}^2-3\times 0\right]$

hence, we get

$=2\times 4=8$

so option $A$ is correct.