[Math] Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

Question

I have a question.

Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$

I don't know how to proceed. Any help would be appreciated.

Answer 1

Use Viete formulas:

$$\alpha\beta = c/a$$$$\alpha + \beta = - b/a$$

Therefore $$\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha^2\beta - 3\alpha\beta^2 = (-b/a)^3 + 3bc/a^2$$