Multiple choice

If α and β are the roots of equation x2 - 2x + 4 = 0, then what is the equation whose roots are α3/β2 and β3/α2?

  1. x2 - 4x + 8 = 0

  2. x2 - 32x + 4 = 0

  3. x2 - 2x + 4 = 0

  4. x2 - 16x + 4 = 0
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C Correct answer
Explanation

Roots of x^2 - 2x + 4 = 0 are alpha, beta. alpha+beta = 2, alpha*beta = 4. New roots are alpha^3/beta^2 and beta^3/alpha^2. Sum = (alpha^5 + beta^5) / (alpha*beta)^2. Product = alpha*beta = 4. Calculating alpha^5 + beta^5 using Newton's sums or expansion gives 32. Sum = 32 / 16 = 2. Equation: x^2 - (sum)x + (product) = x^2 - 2x + 4 = 0.

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