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If $\alpha, \beta$ are the zeroes of the quadratic polynomial $px^2 + qx + r$, then find the value of $\alpha^3\beta + \beta^3\alpha$.
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$px^2 + qx + r$
$\alpha+\beta=-\frac{q}{p}$, $\alpha\beta = \frac{r}{p}$ (1 Mark)
$\alpha^3\beta + \beta^3\alpha$
$\quad = \alpha\beta (\alpha^2 + \beta^2)$ (1/2 Mark)
$\quad = \alpha\beta [(\alpha + \beta)^2 - 2\alpha\beta] = \frac{r}{p} [(-\frac{q}{p})^2 - 2 (\frac{r}{p})]$ (1/2 Mark)
$\quad = \frac{r}{p^3} (q^2 - 2pr)$ (1/2 Mark)
$\alpha+\beta=-\frac{q}{p}$, $\alpha\beta = \frac{r}{p}$ (1 Mark)
$\alpha^3\beta + \beta^3\alpha$
$\quad = \alpha\beta (\alpha^2 + \beta^2)$ (1/2 Mark)
$\quad = \alpha\beta [(\alpha + \beta)^2 - 2\alpha\beta] = \frac{r}{p} [(-\frac{q}{p})^2 - 2 (\frac{r}{p})]$ (1/2 Mark)
$\quad = \frac{r}{p^3} (q^2 - 2pr)$ (1/2 Mark)