Roots of polynomial equations：練習問題

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

The cubic equation \(x^3 - 2x^2 + 3x - 4 = 0\) has roots \(\alpha, \beta, \gamma\). Find a cubic equation whose roots are \(2\alpha, 2\beta, 2\gamma\).

The roots of the cubic equation \(x^3 - px^2 + qx - r = 0\) are in the ratio \(1:2:4\). Which of the following relationships between the coefficients must hold?

The cubic equation \(2x^3 - 4x^2 + 6x - 1 = 0\) has roots \(\alpha, \beta, \gamma\). What is the value of \(\alpha + \beta + \gamma\)?

A cubic equation $2x^3 - 3x^2 + 4x - 1 = 0$ has roots $\alpha, \beta, \gamma$. Find a cubic equation with roots $\alpha+1, \beta+1, \gamma+1$.

The quadratic equation $$2x^2 - 5x + 3 = 0$$ has roots $$\alpha$$ and $$\beta$$. Find the value of $$\alpha + \beta$$ and $$\alpha\beta$$.

The cubic equation $$3x^3 - x^2 + 4x - 2 = 0$$ has roots $$\alpha$$, $$\beta$$, $$\gamma$$. Find the value of $$\sum \alpha^2$$.

The cubic equation \(x^3 + px^2 + qx + r = 0\) has roots \(\alpha, \beta, \gamma\). If \(\alpha, \beta, \gamma\) are in arithmetic progression, show that \(2p^3 + 9pr + 27r = 9pq\).

The roots of the quartic equation \(x^4 + 3x^3 - 2x^2 + 5x - 1 = 0\) are \(\alpha, \beta, \gamma, \delta\). Find the values of \(\sum \alpha\) and \(\sum \alpha\beta\).

Given that \(x=2\) is a root of the cubic equation \(x^3 + kx^2 - 3x - 10 = 0\), find the value of \(k\) and the other two roots of the equation.