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

2 Answers
George C.
May 15, 2015

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.

Dean R.
May 13, 2018

#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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