First slide

Theory of equations

Question

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

Moderate

A

$\frac{d}{a}$
B

$- \frac{d}{a}$
C

$- \frac{b}{a}$
D

$\frac{b}{a}$

Solution

It is given that $x^{2} + x + 1$ is a factor of $a x^{3} + b x^{2} + c x + d$
But $x^{2} + x + 1 = 0$ has its roots as $ω$ and $ω^{2} .$ . So, two imaginary roots of
$a x^{3} + b x^{2} + c x + d = 0$ are $ω$ and $ω^{2} .$ Let the third real root be $α$
Then,
$ω ω^{2} α = (- 1)^{3} \frac{d}{a} \Rightarrow α = - \frac{d}{a}$
Hence, the real root of $a x^{3} + b x^{2} + c x + d = 0$ is $- \frac{d}{a}$ .

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