Quadratic equation: The relationship between solutions and coefficients 1

Question:
The roots of $x^2 - 5x + 3 = 0$ are $\alpha$ and $\beta$. Another quadratic equation $x^2 + px + q =0$ has roots, $\alpha^3$ and $\beta^3$. Find $p$ and $q$.

Answer:
If $\alpha$ and $\beta$ are the roots of a quadratic equation, $ax^2 + bx + c = 0 \quad (a\neq 0)$, the coefficients and roots have the following relationship:
$$\alpha + \beta = -\frac{b}{a}, \\ \alpha \cdot \beta = \frac{c}{a}$$
From the first equation, we have
$$\alpha + \beta = 5, \\ \alpha \cdot \beta = 3$$
From the other equation, they can be expressed as
$$\alpha^3+\beta^3 = -p, \\ (\alpha\beta)^3 = q$$
We can easily have
$$q = 27$$
However, $\alpha^3+\beta^3$ should be modified as follows:
$$\begin{eqnarray*} \alpha^3+\beta^3 &=& (\alpha+\beta)(\alpha^2 -\alpha\beta + \beta^2)  \\                              &=& (\alpha+\beta)(\alpha^2 +2\alpha\beta + \beta^2 - 3\alpha\beta)  \\                             &=& (\alpha+\beta)((\alpha + \beta)^2 - 3\alpha\beta) \end{eqnarray*}$$
Therefore, we obtain
$$p = - (\alpha+\beta)((\alpha + \beta)^2 - 3\alpha\beta) = 5 \cdot (5^2 - 3 \cdot 3) = -80$$

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