Question

Consider the equation \[ x^{2021} + x^{2020} + \cdots + x + 1 = 0. \] Then

• A. all real roots are positive.
• B. exactly one real root is positive.
• C. exactly one real root is negative.
• D. no real root is positive.

Hint:

For equations of the form \(x^n + x^{n-1} + \cdots + x + 1 = 0\), the real roots correspond to the nontrivial roots of unity — i.e., \(x \ne 1.\)

Answer 1

The Correct Option is A, B

Step 1: Simplify the equation.
Multiply both sides by \((x-1)\): \[ (x^{2022} - 1) = 0 \Rightarrow x^{2022} = 1, \quad x \ne 1. \] Hence, all roots are 2022-th roots of unity except \(x=1\).
Step 2: Identify real roots.
The real 2022-th roots of unity are \(x = 1\) and \(x = -1.\) Since \(x=1\) is excluded, the only real root is \(x=-1.\)
Step 3: Conclusion.
Thus, exactly one real root (negative) exists. Hence (C) is correct.