Question

If $x^2+x+1$ is a factor of $a{x}^3+b{x}^2+cx+d$, then real root of $a{x}^3+b{x}^2+cx+d=0$, is:

• A. $-\frac{d}{a}$
• B. $\frac{d}{a}$
• C. $\frac{a}{d}$
• D. None of these

Answer 1

The correct option is A $-\frac{d}{a}$

$x^2+x+1$ is a factor of $a{x}^3+b{x}^2+cx+d$
$\Rightarrow \omega ,{\omega }^2$ are roots of $a{x}^3+b{x}^2+cx+d=0$
Let $\alpha ,\beta ,\gamma$ be the three roots.
$\Rightarrow \alpha \beta \gamma =\frac{-d}{a}$
$\Rightarrow \omega .{\omega }^2.\gamma =\frac{-d}{a}$
$\therefore \frac{-d}{a}$ is the real root of the given equation.
Hence, option A is correct.