1.46: Relationships between roots

(a) Show that $(\alpha+\beta)(\alpha^2-\alpha\beta+\beta^2)=\alpha^3+\beta^3$. (1)

The roots of the equation $2x^2+6x-7=0$ are $\alpha$ and $\beta$, where $\alpha>\beta$. Without solving the equation,

(b) find the value of $\alpha^3+\beta^3$. (4)

(c) show that $\alpha-\beta=\sqrt{23}$. (2)

(d) Hence find the exact value of $\alpha^3-\beta^3$. (2)