$\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \ x^{2}+2 x+3 & 2 x-1 &

Question

$\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}ight|=x A+B$, find A+B?

Professor PC Thomas Classes · Answer

Let $\Delta=\left|\begin{array}{lll}a & b & c \ b & c & a \ c & a & b\end{array}ight|$ Expanding, $\Delta=a\left(b c-a^{2}ight)-b\left(b^{2}-a cight)$\ $+c\left(a b-c^{2}ight)$\ $=-\left[a^{3}+b^{3}+c^{3}-3 a b cight]$\ $=-\left[\begin{array}{r}\frac{1}{2}(a+b+c)\left(2 a^{2}+2 b^{2}+2 c^{2}-2 a bight. \ -2 b c-2 c a)\end{array}ight]$\ $=-\frac{1}{2}(a+b+c)\left[(a-b)^{2}+(b-c)^{2}ight.$\ $\left.+(c-a)^{2}ight]<0$\ $(\because a, b, c$ are +ve and not all equal)\