Question

Find a 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.

• A. $k\{x^3-x^2-x+3\}$, where $k$ is any non-zero real number
• B. $k\{x^3-2x^2-x+3\}$, where $k$ is any non-zero real number
• C. $k\{x^3-4x^2-x+3\}$, where $k$ is any non-zero real number
• D. $k\{x^3-3x^2-x+3\}$, where $k$ is any non-zero real number

Answer 1

Answer: (D)

$k\{x^3-3x^2-x+3\}$, where $k$ is any non-zero real number

Given, $\alpha +\beta +\gamma =3,\alpha \beta +\beta \gamma +\gamma \alpha =-1$ and $\alpha \beta \gamma =-3$
So, polynomial $f\left(x\right)=k\{x^3-x^2(\alpha +\beta +\gamma )+x(\alpha \beta +\beta \gamma +\gamma \alpha )-\alpha \beta \gamma \}$
$f\left(x\right)=k\{x^3-3x^2-x+3\}$,
where $k$ is any non-zero real number