Multiple choice

Let $\alpha , \beta$ be the roots of $x ^ { 2 } + x + 1 = 0$ . the equation whose roots are $\alpha ^ { 19 }$ and $\beta ^ { 16 }$ is

  1. $x ^ { 2 } + x + 1 = 0$
  2. $x ^ { 2 } - x + 1 = 0$
  3. $x ^ { 2 } + x - 1 = 0$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Roots of x^2+x+1=0 are omega and omega^2. omega^3=1. alpha^19 = omega^19 = omega. beta^16 = (omega^2)^16 = omega^32 = omega^2. The roots are omega and omega^2, so the equation is the same.

AI explanation

The roots alpha and beta of the equation x^2 + x + 1 = 0 are the non-real cube roots of unity, meaning alpha^3 = 1 and beta^3 = 1. We can simplify the required roots by evaluating the exponents modulo 3. For the first root, alpha^19 = alpha^(18 + 1) = (alpha^3)^6 * alpha = 1 * alpha = alpha. For the second root, beta^16 = beta^(15 + 1) = (beta^3)^5 * beta = 1 * beta = beta. Since the new roots are exactly alpha and beta, the required equation is identical to the original equation, which is x^2 + x + 1 = 0.

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