Question

Find the cubic polynomial with the sum of zeroes, sum of the products of its zeroes taken two at a time and product of its zeroes as $-4,\frac{1}{2},\frac{-1}{3}$

Answer 1

Let the zeroes be $\alpha ,\beta$ and $\gamma$.
Then, we have
$\alpha +\beta +\gamma =-4$
$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{1}{2}$
$\alpha \beta \gamma =\frac{1}{3}$
Now , required cubic polynomial

$=x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma$

$=x^3-(-4)x^2+\left(\frac{1}{2}\right)x-\left(\frac{1}{3}\right)$

$=\frac{6x^3+24x^2+3x-2}{6}$

So, $6x^3+24x^2+3x-2$ is the required cubic polynomial which satisfy the given conditions.