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Question:

If x² + x + 1 = 0, then x²⁰¹⁸ + x²⁰¹⁹ equals which of the following: 

Explanation:

x² + x + 1 = 0   …(1)

⇒ x² = - x – 1   …(2)

Multiplying x in (2)
⇒ x³ = - x² – x

⇒ x³ = 1   …(3) [From (1)]

Now, 

x²⁰¹⁸ + x²⁰¹⁹ = x²⁰¹⁸(1 + x)

= x²⁰¹⁸ × -x²

= -x²⁰²⁰

= - (x³)⁶⁷³ × x

 = - 1 × x

= - x

∴ x²⁰¹⁸ + x²⁰¹⁹ = - x

Hence, option (a).