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:
Quadratic Equations
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.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root