Question

Find a cubic polynomial having $-3,-2,2$ as its zeroes.

Answer 1

Let the zeroes of the cubic polynomial be
$\alpha =-3,\beta =-2$ and $\gamma =2$
Then, $\alpha +\beta +\gamma =-3+(-2)+2$
$=-3-2+2$
$=-3$
$\alpha \beta +\beta \gamma +\gamma \alpha =(-3)(-2)+(-2)\left(2\right)+\left(2\right)(-3)$
$=6-4-6$
$=-4$
and $\alpha \beta \gamma =(-3)\times (-2)\times 2$
$=6\times 2$
$=12$
Now , required cubic polynomial
$=x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma$
$=x^3-(-3)x^2+(-4)x-12$
$=x^3+3x^2-4x-12$
So, $x^3+3x^2-4x-12$ is the required cubic polynomial which satisfy the given conditions.