Question

Prove that :

$\left|\begin{array}{ccc}b+c& a-b& a\\ c+a& b-c& b\\ a+b& c-a& c\end{array}\right|=3abc-a^3-b-c^3$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{LHS}=\mathrm{\Delta }=\left|b+ca-ba \\ c+ab-cb \\ a+bc-ac\right| \\ \mathrm{\Delta }=\left(b+c\right)\left|b-cb \\ c-ac\right|-\left(a-b\right)\left|c+ab \\ a+bc\right|+a\left|c+ab-c \\ a+bc-a\right|\left[\mathrm{Expanding}\right] \\ =\left(b+c\right)\left\{bc-c^2-bc+ab\right\}-\left(a-b\right)\left\{c^2+ac-ab-b^2\right\}+a\left\{c^2-a^2-ab+ac-b^2+bc\right\} \\ =b{c}^2-c^3+abc-a{c}^2-a^{2}c+a^{2}b+a{b}^2+b{c}^2+abc-a{b}^2-b^3+a{c}^2-a^3-a^{2}c-a{b}^2+abc \\ \Rightarrow \mathrm{\Delta }=3abc-a^3-b^3-c^3\left[\mathrm{Simplyfying}\right] \\ =\mathrm{RHS} \end{gathered}$$