Roots of polynomial equations: แบบฝึกหัด

The roots of the cubic equation \(x^3 - 5x^2 + 7x - 3 = 0\) are \(\alpha\), \(\beta\), and \(\gamma\). Find the value of \(\alpha + \beta + \gamma\).

The equation \(x^4 + 2x^2 + 4x + 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\). Using the substitution \(y = x^2\), find a quartic equation in \(y\) whose roots are \(\alpha^2, \beta^2, \gamma^2, \delta^2\). Hence, find the value of \(\sum \alpha^4\).

The equation \(x^4 + ax^3 + bx^2 + cx + d = 0\) has roots \(\alpha, \beta, \gamma, \delta\). It is given that \(\sum \alpha = 2\), \(\sum \alpha\beta = -1\), \(\sum \alpha\beta\gamma = -4\), and \(\alpha\beta\gamma\delta = 1\). Find the value of \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} + \frac{1}{\delta^2}\).

The roots of the cubic equation \(x^3 - 3x^2 + 4x - 6 = 0\) are \(\alpha\), \(\beta\), and \(\gamma\).
Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).

Given that \(\alpha\), \(\beta\), and \(\gamma\) are the roots of the cubic equation \(x^3 + px^2 + qx + r = 0\), find the value of \(\alpha^2 + \beta^2 + \gamma^2\) in terms of the coefficients.

The roots of the quartic equation \(2x^4 - 8x^3 + 5x^2 - x + 6 = 0\) are \(p\), \(q\), \(r\), and \(s\). Find the value of \(\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s}\).

The roots of the cubic equation \(x^3 + 3x - 2 = 0\) are \(\alpha\), \(\beta\), and \(\gamma\). Use the substitution \(u = x + 2\) to find a cubic equation in \(u\) whose roots are \(\alpha + 2\), \(\beta + 2\), and \(\gamma + 2\).

The quartic equation \(x^4 - 3x^3 + 2x^2 - x + 5 = 0\) has roots \(\alpha, \beta, \gamma, \delta\). Find the value of \(\sum \alpha^2 \beta^2\).

The quadratic equation \( x^2 + px + q = 0 \) has roots \( \alpha \) and \( \beta \). Given that \( \alpha + \beta = 7 \) and \( \alpha^2 + \beta^2 = 25 \), find the values of the constants p and q.

Given that the roots of \( x^3 - 6x^2 + kx - 8 = 0 \) are in geometric progression, find:
(a) the value of the constant k,
(b) the three roots of the equation.