1.26: Past-paper question 5

(a) Show that $\displaystyle (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = \alpha^3 + \beta^3$

The quadratic equation $\displaystyle 2x^2 - 6x - 7 = 0$ has roots $\displaystyle \alpha$ and $\displaystyle \beta$

Without solving the equation

(b) form a quadratic equation, with integer coefficients, which has roots $\displaystyle \frac{\alpha^2}{\beta}$ and $\displaystyle \frac{\beta^2}{\alpha}$