Is this factorable?

x is a natural number
a,b are real numbers

a^2 + b^2
a^4 + b^4

and more generally:
a^(2x) + b^(2x)

Because, every polynomials should have real roots, right?

Question

Is this factorable?

x is a natural number
a,b are real numbers

a^2 + b^2
a^4 + b^4

and more generally:
a^(2x) + b^(2x)

Because, every polynomials should have real roots, right?

mayankdevnani · Answer

@pourushgupta

hartnn · Answer

nopes, polynomials can have imaginary roots.
a^2+b^2 can be factored as (a+ib)(a-ib)

hartnn · Answer

a^4+b^4 can be factored as $a^4+2a^2b^2+b^4-2a^2b^2 = (a^2+b^2)^2-(\sqrt 2ab)^2$
then using x^2-y^2 formula

anonymous · Answer

what about the general case?

hartnn · Answer

i cannot think of any way to factor that general case, but u can find the root(if u consider that as equation in x), by equating it to 0, it will be imaginary root.

unklerhaukus · Answer

$$(a^2 + b^2)=(a+ib)(a-ib)$$
$$(a^4 + b^4)=(A^2+B^2)$$$$=(A+iB)(A-iB)=(a^2+ib^2)(a^2-ib^2)$$