Sophie Germain's Identity/Proof 1
Jump to navigation
Jump to search
Theorem
- $x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$
Proof
\(\ds \) | \(\) | \(\ds \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^4 + x^2 \cdot 2 y^2 - x^2 \cdot 2 x y + x^2 \cdot 2 y^2 + 4 y^2 - 2 y^2 \cdot 2 x y + x^2 \cdot 2 x y + 2 y^2 \cdot 2 x y - 2 x y \cdot 2 x y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^4 + 4 y^4\) | gathering up terms and cancelling |
$\blacksquare$