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Let alpha and beta be the roots of the equation x² - x + 1 = 0. Find the value of 3*alpha³ - 3*alpha² + 2*beta³ - 2*beta² + 11*alpha.

1. 33
2. -33
3. 22
4. -22

Correct answer: 33

Solution

Since alpha and beta are roots of x² - x + 1 = 0, we have alpha² = alpha - 1 and alpha³ = -1. Substituting into the expression gives 3(-1) - 3(alpha-1) + 2(-1) - 2(beta-1) + 11*alpha. Using alpha + beta = 1 and collecting terms yields the answer.

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