Prove that:


Prove that:

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


Let $\mathrm{LHS}=\Delta=\mid b+c a-b \quad a$

$c+a b-c b$

$a+b c-a c$

$\Delta=(b+c) \mid b-c \quad b$

$c-a c|-(a-b)| c+a b$

$a+b c|+a| c+a b-c$

$a+b c-a \mid \quad[$ Expanding $]$

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

$=b c^{2}-c^{3}+a b c-a c^{2}-a^{2} c+a^{2} b+a b^{2}+b c^{2}+a b c-a b^{2}-b^{3}+a c^{2}-a^{3}-a^{2} c-a b^{2}+a b c$

$\Rightarrow \Delta=3 a b c-a^{3}-b^{3}-c^{3} \quad[$ Simplyfying $]$

$=$ RHS

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