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
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
- -4
- -1
- 1
- 4
The Correct Option is B
Solution and Explanation
x4 + x3 + x2 + x + 1 = 0 or \(\frac{x^5-1}{x-1}\) = 0.
So roots are e\(^{\frac{i2\pi}{5}}\), e\(^{\frac{i4\pi}{5}}\), e\(^{\frac{i6\pi}{5}}\), e\(^{\frac{i8\pi}{5}}\), i.e. α,β,γ and δ
From properties of nth root of unity
\(1^{2021}+α^{2021}+β^{2021}+γ^{2021}+δ^{2021}\)=0
⇒ \(α^{2021}+β^{2021}+γ^{2021}+δ^{2021}=-1\)
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Concepts Used:
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A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
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Two important points to keep in mind are:
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- Taking the square root