How do you factor $a^2 + b^2$ ?

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How do you factor $a^2 + b^2$ ?

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Answer 1 · 2015-05-15T22:15:48

Whereas $a^2-b^2 = (a+b)(a-b)$ is very simple, to factor $a^2+b^2$ requires the use of complex numbers.

If $i = sqrt(-1)$ then

$(a+ib)(a-ib)$

$=a^2+iab-iab-i^2b$

$= a-i^2b$

$= a^2-(-1)b^2$

$= a^2 + b^2$

So $a^2+b^2 = (a+ib)(a-ib)$ , but there is no other factoring with real number coefficients.

Answer 2 · 2018-05-13T14:15:59

$a^2+b^2$ doesn't have a nice factorization over the reals, but over the complex numbers it's the squared magnitude of $a+bi,$ which gives the factorization

$(a+bi)(a-bi)=a^2+b^2$

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