Multiple choice

If α and β are the roots of the equation x2 - 2x + 4 = 0, then (α5 + β5) is equal to

  1. 243

  2. 81

  3. 64

  4. 32
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The roots satisfy x^2 = 2x - 4. Then x^3 = 2x^2 - 4x = 2(2x - 4) - 4x = -8. Thus, x^5 = -8x^2 = -8(2x - 4) = -16x + 32. Summing for alpha and beta: (alpha^5 + beta^5) = -16(alpha + beta) + 64. Since alpha + beta = 2, the result is -16(2) + 64 = 32.

AI explanation

For the equation x^2 - 2x + 4 = 0, the sum of the roots (alpha + beta) is 2 and the product of the roots (alpha beta) is 4. We find (alpha + beta)^2 = 2^2 = 4, so alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha beta) = 4 - 8 = -4. Next, (alpha^2 + beta^2)(alpha + beta) = (-4)(2) = -8, which expands to alpha^3 + beta^3 + alpha beta (alpha + beta) = -8. Substituting the known values gives alpha^3 + beta^3 + (4)(2) = -8, so alpha^3 + beta^3 = -16. Finally, (alpha^3 + beta^3)(alpha^2 + beta^2) = (-16)(-4) = 64, which expands to alpha^5 + beta^5 + alpha^2 beta^2 (alpha + beta) = 64. Substituting the known values yields alpha^5 + beta^5 + (16)(2) = 64, so alpha^5 + beta^5 = 32.

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