<s>[INST] I cannot find the last factor of this expression? I'm supposed to factor $x^8-y^8$ (the exponents are 8 for both if it is too difficult to see) as completely as possible. It is easy to factor this to $(x+y)(x-y)(x^2+y^2)(x^4+y^4)$. However, the book says "(Hint: there are 5 factors. Note that we sat real coefficients, not just integers)". What is this elusive 5th factor? Thanks!
 [/INST] Another approach: if you know that
$$x^4+1=(x^2+\sqrt 2x+1)(x^2-\sqrt 2x+1)$$
then you can do as follows:
$$x^4+y^4=y^4\left(\frac{x^4}{y^4}+1\right)=y^4\left(\frac{x^2}{y^2}+\sqrt 2\frac xy+1\right)\left(\frac{x^2}{y^2}-\sqrt 2\frac xy+1\right)=$$
$$=\left(x^2+\sqrt 2xy+y^2\right)\left(x^2-\sqrt 2xy+y^2\right)$$
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