przekształcenie wielomianu

[olussskaaa]

Przedstaw wielomian \(\displaystyle{ x^8+x^4+1}\) w postaci iloczynu czterech wielomianów stopnia 2.

[Brzytwa]

\(\displaystyle{ x^8+x^4+1=x^{8}+2x^{4}+1-x^{4}=(x^{4}+1)^{2}-x^{4}=(x^{4}+x^{2}+1)(x^{4}-x^{2}+1)=(x^{4}+2x^{2}+1-x^{2})(x^{4}+2x^{2}+1-3x^{2})=((x^{2}+1)^{2}-x^{2})((x^{2}+1)^{2}-3x^{2})=(x^{2}+x+1)(x^{2}-x+1)(x^{2}+\sqrt{3}x+1)(x^{2}-\sqrt{3}x+1)}\)

[Mersenne]

\(\displaystyle{ x^{8}+x^{4}+1=(x^{4}+1)^{2}-x^{4}=(x^{4}+1)^{2}-(x^{2})^{2}=(x^{4}+1-x^{2})(x^{4}+1+x^{2})=}\)

\(\displaystyle{ =[(x^{2}+1)^{2}-3x^{2}]\cdot [(x^{2}+1)^{2}-x^{2}]=[(x^{2}+1)^{2}-(\sqrt{3}x)^{2}] [(x^{2}+1)^{2}-x^{2}]=}\)

\(\displaystyle{ =(x^{2}+1-\sqrt{3}x)(x^{2}+1+\sqrt{3}x)(x^{2}+1-x)(x^{2}+1+x)}\)