1.35: Relationships between roots

The equation $3x^2-4x+6=0$ has roots $\alpha$ and $\beta$.

(a) Without solving the equation, write down

(i) the value of $\alpha+\beta$

(ii) the value of $\alpha\beta$ (2)

(b) Without solving the equation, show that $\alpha^3+\beta^3=-\frac{152}{27}$. (3)

(c) Form a quadratic equation, with integer coefficients, that has roots $\frac{\alpha}{\beta^2}$ and $\frac{\beta}{\alpha^2}$. (5)