Question

Fill all the roots of the equation.

$x^6$ - $x^5$ + $x^4$ - $x^3$ + $x^2$ - x + 1 = 0 1 + x $\ne 0$

• A. + ; n = 0,1,2,3,4,5,6
• B. + ; n = 0,1,2,3,4,5,6
• C. + ; n = 0,1,2,3,4,5,6
• D. + ; n = 0,1,2,3,4,5,6

Answer 1

The correct option is B

+ ; n = 0,1,2,3,4,5,6

Given 1 - x + $x^2$ - $x^3$ + $x^4$ - $x^5$ + $x^6$ = 0

We see that given series is a GP with common ratio (-x)

Sum of the GP

$\frac{1(1-(-x{)}^7)}{1-(-x)}$ = 0 Where 1 + x $\ne 0$

1 + $x^7$ = 0

$x^7$ = -1

x = $(-1{)}^{\frac{1}{7}}$ = $(cos\pi +isin\pi {)}^{\frac{1}{7}}$

= $cos\frac{(2n+1)\pi }{7}$ + $isin\frac{(2n+1)\pi }{7}$;

n = 0,1,2,3,4,5,6