Question

If the sum of two of the roots of $x^3-5x^2-4x+20=0$ is zero, then the roots are

• A. $2,-2,4$
• B. $2,-2,3$
• C. $2,-2,5$
• D. $2,-2,6$

Answer 1

The correct option is C $2,-2,5$

Consider the roots to be $\alpha ,\beta ,\gamma$
Then general form will be
$x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma =0$
Now it is given that $\alpha +\beta =0$
Therefore $\gamma (\alpha +\beta )=0$
$\alpha \gamma +\beta \gamma =0$ and
Substituting in the equation, we get
$x^3-\gamma \left(x^2\right)+\left(\alpha \beta \right)x-\alpha \beta \gamma =0$
Now $\alpha +\beta =0$
$\Rightarrow \alpha =-\beta$
Substituting in the above equation, we get
$x^3-\gamma \left(x^2\right)-{\alpha }^2\left(x\right)+{\alpha }^2\gamma =0$
Comparing with $x^3-5x^2-4x+20=0$, we get

$\gamma =5$ and ${\alpha }^2=4$
$\alpha =\pm 2$, hence $\beta =\mp 2$.
Therefore $(\alpha ,\beta ,\gamma )=(2,-2,5)$ or $(-2,2,5)$.