Multiple choice

Let α and β be the roots of the equation x2 + px + q = 0. If α3 and β3 are the roots of the equation x2 + mx + n = 0, then what is the value of m + n?

  1. p3 + q3 + pq

  2. p3 + q3 - pq

  3. p3 + q3 + 3pq

  4. p3 + q3 - 3pq
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D Correct answer
Explanation

Roots alpha, beta satisfy alpha+beta = -p, alpha*beta = q. New roots alpha^3, beta^3 satisfy sum = -m, product = n. alpha^3 + beta^3 = (alpha+beta)^3 - 3*alpha*beta*(alpha+beta) = (-p)^3 - 3*q*(-p) = -p^3 + 3pq. So m = p^3 - 3pq. Product = (alpha*beta)^3 = q^3 = n. m+n = p^3 - 3pq + q^3.

AI explanation

Using Vieta's formulas for x^2 + px + q = 0, the sum of the roots is p = -(alpha + beta) and the product is q = alpha*beta. For the new equation x^2 + mx + n = 0 with roots alpha^3 and beta^3, the sum of the roots is -m = alpha^3 + beta^3 and the product is n = alpha^3*beta^3. Using the sum of cubes identity, alpha^3 + beta^3 = (alpha + beta)^3 - 3*alpha*beta(alpha + beta), we substitute the known values to get -m = (-p)^3 - 3q(-p) = -p^3 + 3pq, so m = p^3 - 3pq. We also have n = (alpha*beta)^3 = q^3, and adding these together gives m + n = p^3 + q^3 - 3pq.

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