If (alpha) and (beta) are the roots of (3x^{2} + 5x + 1 = 0), evaluate (27(alpha^{3} + beta^{3})).
Explanation
(3x^{2} + 5x + 1 = 0)
(x^{2} + frac{5}{3}x + frac{1}{3} = 0)
(alpha beta = frac{c}{a})
= (frac{1}{3})
(alpha + beta = frac{-b}{a})
= (-frac{5}{3})
((alpha + beta)^{3} = alpha^{3} + 3alpha^{2} beta + 3alpha beta^{2} + beta^{3})
(therefore alpha^{3} + beta^{3} = (alpha + beta)^{3} – 3alpha^{2} beta – 3alpha beta^{2})
= ((alpha + beta)^{3} – 3alpha beta (alpha + beta))
= ((-frac{5}{3})^{3} – 3(frac{1}{3})(-frac{5}{3}))
= ((-frac{125}{27}) + frac{5}{3})
(therefore (alpha^{3} + beta^{3}) = -frac{80}{27})
(27(alpha^{3} + beta^{3}) = 27(-frac{80}{27}))
= -80.