Question

if 1,${\alpha }_1$,${\alpha }_2$,${\alpha }_3$................${\alpha }_7$ are the roots of the equation $x^8$ - 1 = 0.Find of the value of ${\sum }_{i=1}^7({\alpha }_i{)}^2$

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Answer 1

Given $x^8$ = 1

Sum of the roots

1 + ${\alpha }_1$ + ${\alpha }_2$ + ${\alpha }_3$ + ................${\alpha }_7$ = 0

${\sum }_{i=1}^7({\alpha }_i{)}^2$

= $({\alpha }_1{)}^2$ + $({\alpha }_2{)}^2$ + $({\alpha }_3{)}^2$ + ................$({\alpha }_7{)}^2$

= $\underset{}{(1{)}^2+({\alpha }_1{)}^2+({\alpha }_2{)}^2+.......................({\alpha }_7{)}^2}$ - 1
0 $\left\{\begin{array}{c}{1}^p+({\alpha }_1{)}^p+({\alpha }_2{)}^p+.......({\alpha }_7{)}^p=0\\ ifpisnotamultipleofn\end{array}\right\}$

= 0 - 1 = -1