If α, β, γ, δ are the roots of the equation x⁴+x³+x²+x+1=0, then α²⁰²¹+β²⁰²¹+γ²⁰²¹+δ²⁰²¹ is equal to :

Solution

Maths sir · 4 steps

Multiply by (x-1) and these four roots become fifth roots of unity, so the big power collapses.

1. Spot the pattern. Since (x-1)(x⁴+x³+x²+x+1) = x⁵ - 1, every root of the given equation satisfies x⁵ = 1. The roots α, β, γ, δ are the four fifth roots of unity other than 1.
2. Reduce the power. Divide 2021 by 5: 2021 = 5 × 404 + 1. So for each root, x²⁰²¹ = (x⁵)⁴⁰⁴ · x = 1 · x = x.

   α²⁰²¹+β²⁰²¹+γ²⁰²¹+δ²⁰²¹ = α+β+γ+δ

3. Sum of roots. For x⁴ + x³ + x² + x + 1 = 0, the sum of the roots is -(coefficient of x³)/(coefficient of x⁴) = -1/1 = -1.
4. So. The required sum is α+β+γ+δ = -1, which is option (b).

The idea to keep. When roots satisfy xⁿ = 1, reduce any huge power by taking the exponent modulo n. Then Vieta's formulas usually finish the job.

Answer from the NTA final answer key. Solution by PrepWiser.