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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