factorize-inside-R-x-the-polynomial-p-x-x-8-1-

Question Number 29169 by abdo imad last updated on 04/Feb/18

f a c t o r i z e i n s i d e R [x] t h e p o l y n o m i a l

p (x) = x^{8} - 1 .

Commented by abdo imad last updated on 06/Feb/18

p (x) = {(x^{4})}^{2} - 1 = (x^{4} - 1) (x^{4} + 1) = (x^{2} - 1) (x^{2} + 1) (x^{4} + 1)

= (x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1) b u t

x^{4} + 1 = {(x^{2} + 1)}^{2} - 2 x^{2} = (x^{2} + 1 - \sqrt{} 2 x) (x^{2} + 1 + \sqrt{} 2 x) s o

p (x) = (x - 1) (x + 1) (x^{2} + 1) (x^{2} - \sqrt{2} x + 1) (x^{2} + \sqrt{2} x + 1) .