Question:
Find the cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, −1 and − 3 respectively.
Solution:
If $\alpha, \beta$ and $\gamma$ are the zeros of a cubic polynomial $f(x)$, then
$f(x)=k\left\{x^{3}-(\alpha+\beta+\gamma) x^{2}+(\alpha \beta+\beta \gamma+\gamma \alpha) x-\alpha \beta \gamma\right\}$ where $k$ is any non-zero real number.
Here,
$\alpha+\beta+\gamma=3$
$\alpha \beta+\beta \gamma+\gamma \alpha=-1$
$\alpha \beta \gamma=-3$
Therefore
$f(x)=k\left\{x^{3}-(3) x^{2}+(-1) x-(-3)\right\}$
$f(x)=k\left\{x^{3}-3 x^{2}-1 x+3\right\}$
Hence, cubic polynomial is $f(x)=k\left\{x^{3}-3 x^{2}-1 x+3\right\}$, where $k$ is any non-zero real number.