Express $x_1^3 + x_2^3$ in terms of the coefficients of the equation $x^2 + px + q = 0$, where $x_1$ and $x_2$ are the roots of the equation.
Given quadratic equation: $\;$ $x^2 + px + q = 0$
Roots of the given quadratic equation are $\;$ $x_1$ $\;$ and $x_2$.
Sum of roots $= x_1 + x_2 = -p$ $\;\;\; \cdots \; (1a)$
Product of roots $= x_1 \cdot x_2 = q$ $\;\;\; \cdots \; (1b)$
Now,
$\begin{aligned}
x_1^3 + x_2^3 & = \left(x_1 + x_2\right) \left[x_1^2 - x_1 \cdot x_2 + x_2^2\right] \\\\
& = \left(x_1 + x_2\right) \left[\left(x_1 + x_2\right)^2 - 2x_1 \cdot x_2 - x_1 \cdot x_2\right] \\\\
& = \left(x_1 + x_2\right) \left[\left(x_1 + x_2\right)^2 - 3 x_1 \cdot x_2\right] \;\;\; \cdots \; (2)
\end{aligned}$
In view of equations $(1a)$ and $(1b)$, equation $(2)$ becomes
$\begin{aligned}
x_1^3 + x_2^3 & = \left(-p\right) \left[\left(-p\right)^2 - 3q\right] \\\\
& = 3pq - p^3
\end{aligned}$