If x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0, then, find the value of x⁴² + x⁸⁴.

Question

Question

Answer 1

Given:

x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0 ----(1)

Calculation:

Multiplying by x, we have –

⇒ x⁷ +x⁶ + x⁵ + x⁴ + x³ + x² + x = 0 ----(2)

Equation (2) – (1), we have –

⇒ x7 – 1 = 0

⇒ x⁷ = 1

According to question –

The value of x⁴² + x⁸⁴= \({\left( {{x^7}} \right)^6} + {\left( {{x^7}} \right)^{12}} = {\left( 1 \right)^6} + {\left( 1 \right)^{12}} = 2\)