If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 5x^2 - 7x - 1 = 0$,then find the equation whose roots are $\alpha\beta, \beta\gamma, \gamma\alpha$.

If $\alpha$ and $\beta$ are the roots of the equation $4x^2 - \sqrt{13}x - 7 = 0$,then what is the value of $|\alpha - \beta|$?

If one root of $ax^2 + bx + c = 0$ is the square of the other,then the value of $b^3 + ac^2 + a^2c$ is

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$,with $\alpha>\beta$. For all positive integers $n$,define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$ and $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following options is/are correct?
$(1)$ $\sum_{i=1}^{n} a_i = a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
$(4)$ $b_n = \alpha^n+\beta^n$ for all $n \geq 1$

If one root of $5x^2 + 13x + k = 0$ is the reciprocal of the other,then $k = $?

Let $r_1, r_2, r_3$ be roots of the equation $x^3 - 2x^2 + 4x + 5074 = 0$. Then the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is