# Find the cubic polynomial with the sum,

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.