Question

Using the factor theorem it is found that $b+c,c+a$ and $a+b$ are three factors of the determinant $\left|\begin{array}{ccc}-2a& a+b& a+c\\ b+a& -2b& b+c\\ c+a& c+b& -2c\end{array}\right|.$ The other factor in the value of the determinant is

• A. $4$
• B. $2$
• C. $a+b+c$
• D. none of these

Answer 1

Answer: (A)

$4$

$\left|\begin{array}{ccc}-2a& a+b& a+c\\ b+a& -2b& b+c\\ c+a& c+b& -2c\end{array}\right|$

$-2a(4b-{(b+c)}^2)-(a+b)[-2c(b+a)-(b+c)(c+a)]$

$+(a+c)[(b+a)(+b)+2b(a+c)]$

$=4(b+c)(c+a)(a+b)$

$\therefore$Other factor is $4$