Evaluate the following : x^6 + x^4 + x^2 + 1, when x = 1+i/√2

Question

Evaluate the following : x⁶ + x⁴ + x² + 1, when x = \(\frac{1+i}{\sqrt{2}}\)

Answer 1

Given: 

⇒ x = \(\frac{1+i}{\sqrt{2}}\)

⇒ x⁶ + x⁴ + x² + 1 

⇒ x⁴ (x²+1)+1(x²+1) 

⇒ (x⁴ +1)(x²+1)

⇒ (-1+1)(i+1) 

⇒ (0)(i+1) 

⇒ 0

∴ x⁶ + x⁴ + x² + 1 = 0