Find a cubic polynomial with the sum of its roots, the sum of the products of its roots taken two at a time, and the product of its roots as 2, -7, and -14, respectively.

Suppose, $\alpha ,\mathit{ \beta },\gamma$ are the zeroes of a cubic polynomial.

Given:
The sum of roots, $\alpha +\beta +\gamma =2$
The sum of the product of zeroes when taken 2 at a time,

$\alpha \beta +\beta \gamma +\gamma \alpha =-7$

Also,
The product of zeroes $\mathit{=}\mathit{\alpha \beta \gamma }\mathit{=}\mathit{-}14$

So, the polynomial will be

$=x^3-\left(\alpha +\beta +\gamma \right)x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\left(\mathit{\alpha \beta \gamma }\right)$

$=x^3-2x^2-7x+14$

So, the required cubic polynomial is: ${\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathit{x}\mathbf{+}\mathbf{14}$