Prove that-1 b - cb 2- c 21 c - ac 2- a 21 a - b a 2- b 2- a - b b - c c - a

Let #160;LHS #160;= #8710;= #160;1 #160; #160; #160; #160;b+c #160; #160; #160; #160; #160;b2+c21 #160; #160; #160; #160;c+a ...

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Algebra of Complex Numbers

Prove that:1 ...

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Prove that :

$|\begin{matrix} 1 & b + c & b^{2} + c^{2} \\ 1 & c + a & c^{2} + a^{2} \\ 1 & a + b & a^{2} + b^{2} \end{matrix}| = (a - b) (b - c) (c - a)$

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(\frac{x^{a}}{x^{- b}})^{a^{2} - a b + b^{2}} \times (\frac{x^{b}}{x^{- c}})^{b^{2} - b c + c^{2}} \times (\frac{x^{c}}{x^{- a}})^{c^{2} - c a + a^{2}} =

x^{2 (a^{3} + b^{3} + c^{3})}

∣ ∣ ∣ ∣ \begin{matrix} b + c & c + a & a + b c + a & a + b & b + c a + b & b + c & c + a \end{matrix} ∣ ∣ ∣ ∣ == 2 (a + b + c) (a b + b c + c a - a^{2} - b^{2} - c^{2})

|\begin{matrix} 1 & a^{2} + b c & a^{3} \\ 1 & b^{2} + c a & b^{3} \\ 1 & c^{2} + a b & c^{3} \end{matrix}| = - (a - b) (b - c) (c - a) (a^{2} + b^{2} + c^{2})

∣ ∣ ∣ ∣ \begin{matrix} a & b - c & c + b a + c & b & c - a a - b & b + a & c \end{matrix} ∣ ∣ ∣ ∣ = (a + b + c) (a^{2} + b^{2} + c^{2})

|\begin{matrix} {(b + c)}^{2} & a^{2} & b c \\ {(c + a)}^{2} & b^{2} & c a \\ {(a + b)}^{2} & c^{2} & a b \end{matrix}| = (a - b) (b - c) (c - a) (a + b + c) (a^{2} + b^{2} + c^{2})

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