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Sum of Coefficients of All Terms

If a + b + ...

Question

If $a + b + c = 0$ , prove that identities $3 (a^{2} + b^{2} + c^{2}) (a^{5} + b^{5} + c^{5}) = 5 (a^{3} + b^{3} + c^{3}) (a^{4} + b^{4} + c^{4})$ .

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Solution

$a^{2} + b^{2} + c^{2} = - 2 q, a^{5} + b^{5} + c^{5} = - 5 q r$
$a^{3} + b^{3} + c^{3} = 3 r, a^{4} + b^{4} + c^{4} = 2 q^{2}$
LHS;-
$3 (a^{2} + b^{2} + c^{2}) (a^{5} + b^{5} + c^{5})$
$= 3 (- 2 q) (- 5 q r) = 30 q^{2} r$
RHS;-
$5 (a^{3} + b^{3} + c^{3}) (a^{4} + b^{4} + c^{4})$
$= 5 (3 r) (2 q^{2}) = 30 q^{2} r$
Hence proved.

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