Factorise the polynomial x^8 + 16?

Question

Step by step please

Factorise it into polynomial factors (linear or quadratic) which have real coefficients

Answer 1

We can't factorise it over Q, but we can factorise it
over the reals.
Write it as x^8 + 8x^4 + 16 - 8x^4
= (x^4+4)² - 8x^4 = (x^4 + √8x ^2+ 4)(x^4 - √8x^2 + 4)
Now go to work on the other 2 factors:
x^4 + √8 x^2 + 4 = x^4 + 4x^2 + 4 -(4-√8)x^2
which factors as
(x^2+ √(4 - √8) x + 2)(x^2- √(4 - √8) x + 2).
The results for the other factor are the same
except we get 4 + √8 in each factor instead of 4 -√8.
Putting it all together gives the result you seek.
Unfortunately, each of these quadratic factors
has negative discriminant, so we can go no further.

Answer 2

This polynomial cannot be factorized further.

Answer 3

Unfortunately, you can't factorize this any further.