Răspuns:
9
Explicație pas cu pas:
Desfacem in produs expresia [tex] \alpha^3=1 [/tex].
[tex]\alpha ^3=1\\\alpha^3-1=0\\(\alpha-1)(\alpha^2+\alpha+1)=0\\Deci, \alpha^2+\alpha+1=0~si~\alpha^2+\alpha=-1[/tex]
Stiind ca inmultirea este asociativa, grupam parantezele doua cate doua si facem calculele ce se impun.
Produsul primelor doua paranteze este:
[tex](1-\alpha)(1-\alpha^2)=1-\alpha^2-\alpha+\alpha^3=1+1-(\alpha^2+\alpha)=2-(-1)=3[/tex]
Produsul ultimelor doua paranteze este:
[tex](1-\alpha^4)(1-\alpha^5)=1-\alpha^5-\alpha^4+\alpha^9=1+(\alpha^3)^3-\alpha^3(\alpha^2+\alpha)=2-\alpha^3*(-1)=2+\alpha^3=2+1=3[/tex]
Finalizam:
[tex](1-\alpha)(1-\alpha^2)(1-\alpha^4)(1-\alpha^5)=3*3=9 [/tex]
