The cubic equation \(x^3 + bx^2 + cx + d = 0\) has roots \(\alpha, \beta, \gamma\).
Find the values of \(b, c\) and \(d\).
Solution
Given \(\alpha + \beta + \gamma = 3\), we have \(b = - (\alpha + \beta + \gamma) = -3\).
Using the formula for the sum of squares: \(\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)\),
Using the formula for the sum of cubes: \(\alpha^3 + \beta^3 + \gamma^3 = 3(\alpha\beta\gamma) + (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha)\),
\(6 = 3(\alpha\beta\gamma) + 3(5 - 1)\).
Solving gives \(\alpha\beta\gamma = 1\), so \(d = 1\).
