Question

Let α be a root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² – α³⁰³³ is equal to ______.

Options

• 1

• α

• 1 + α

• 1 + 2α

Answer 1

Let α be a root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² – α³⁰³³ is equal to 1.

Explanation:

Given: 1 + x² + x⁴ = 0

⇒ x⁴ – x² + x² + x² + x – x + 1 = 0

⇒ (x² – x)(x² + x) + (x² – x) + (x² + x + 1) = 0

⇒ (x² – x) (x² + x + 1) + (x² + x + 1) = 0

⇒ (x² + x + 1)(x² – x + 1) = 0

⇒ x² + x + 1 = 0 or x² – x² + 1 = 0

⇒ x = ω, ω² or x = ω, ω², where ω is a cube root of unity.

⇒ α = ω

Now, α¹⁰¹¹ + α²⁰²² – α³⁰³³ = ω¹⁰¹¹ + ω²⁰²² – ω³⁰³³

= (ω³)³³⁷ + (ω³)⁶⁷⁴ – (ω³)¹⁰¹¹

= 1 + 1 – 1   ...{∵ ω³ = 1}

= 1