Sophie Germain's Identity

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Theorem

For any two numbers $x$ and $y$:

$x^4 + 4y^4 = \left({x^2 + 2y^2 + 2xy}\right) \left({x^2 + 2y^2 - 2xy}\right)$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \left({x^2 + 2y^2 + 2xy}\right) \left({x^2 + 2y^2 - 2xy}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x^4 + x^2.2y^2 - x^2.2xy + x^2.2y^2 + 4 y^2 - 2y^2.2xy + x^2.2xy + 2y^2.2xy - 2xy.2xy$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x^4 + 4 y^4$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by gathering up terms and cancelling

$\blacksquare$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \left({x^2 + 2y^2 + 2xy}\right) \left({x^2 + 2y^2 - 2xy}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({x^2 + 2y^2}\right)^2 - \left({2xy}\right)^2$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Difference of Two Squares
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x^4 + 2.x^2.2y^2 + 4 y^2 - 2xy.2xy$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x^4 + 4 y^4$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by gathering up terms and cancelling

$\blacksquare$

This entry was named for Sophie Germain.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \left({x^2 + 2y^2 + 2xy}\right) \left({x^2 + 2y^2 - 2xy}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x^4 + x^2.2y^2 - x^2.2xy + x^2.2y^2 + 4 y^2 - 2y^2.2xy + x^2.2xy + 2y^2.2xy - 2xy.2xy\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x^4 + 4 y^4\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by gathering up terms and cancelling

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \left({x^2 + 2y^2 + 2xy}\right) \left({x^2 + 2y^2 - 2xy}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x^2 + 2y^2}\right)^2 - \left({2xy}\right)^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Difference of Two Squares
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x^4 + 2.x^2.2y^2 + 4 y^2 - 2xy.2xy\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x^4 + 4 y^4\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by gathering up terms and cancelling