1.40: Relationships between roots

It is given that $\alpha$ and $\beta$ are such that $\alpha+\beta=-\dfrac52$ and $\alpha\beta=-5$.

(a) Form a quadratic equation with integer coefficients that has roots $\alpha$ and $\beta$. (2)

Without solving the equation found in part (a)

(b) find the value of

(i) $\alpha^2+\beta^2$

(ii) $\alpha^3+\beta^3$ (5)

(c) Hence form a quadratic equation with integer coefficients that has roots