sophie−Germain identity a^4 +4b^4 =((a+b)^2 +b^2 )((a−b)^2 +b^2 )


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Question Number 70135 by petrochengula last updated on 01/Oct/19

$s o p h i e - G e r m a i n i d e n t i t y$ $a^{4} + 4 b^{4} = ({(a + b)}^{2} + b^{2}) ({(a - b)}^{2} + b^{2})$
Commented by mathmax by abdo last updated on 02/Oct/19
${(a+b)^2 +b^2 ){(a−b)^2 +b^2 }= =(a^2 +2ab +2b^2 )(a^2 −2ab +2b^2 ) =(a^2 +2b^2 +2ab)(a^2 +2b^2 −2ab) =(a^2 +2b^2 )^2 −4a^2 b^2 =a^4 +4a^2 b^2 +4b^4 −4a^2 b^2 =a^4 +4b^4$
${{(a + b)}^{2} + b^{2}) {{(a - b)}^{2} + b^{2}} =$ $= (a^{2} + 2 a b + 2 b^{2}) (a^{2} - 2 a b + 2 b^{2})$ $= (a^{2} + 2 b^{2} + 2 a b) (a^{2} + 2 b^{2} - 2 a b)$ $= {(a^{2} + 2 b^{2})}^{2} - 4 a^{2} b^{2} = a^{4} + 4 a^{2} b^{2} + 4 b^{4} - 4 a^{2} b^{2}$ $= a^{4} + 4 b^{4}$
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