First slide

Theory of equations

Question

If $x^{2} + x + 1$ is a factor of ${ax}^{3} + {bx}^{2} + cx + d$ the real roots of ${ax}^{3} + {bx}^{2} + cx + d = 0$ is

Moderate

A

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

$\frac{d}{a}$
C

$\frac{a}{d}$
D

None of these

Solution

We know that, x ²+ x + 1 is a factor of ${ax}^{3} + {bx}^{2} + cx + d$
Hence, roots of x ²+ x + 1 are also roots of
${ax}^{3} + {bx}^{2} + cx + d$ . Since, $ω and ω^{2}$
$(where ω = - \frac{1}{2} + \frac{3 i}{2})$ are two complex roots x ²+ x + 1
Therefore, $ω and ω^{2}$ are two complex roots of ${ax}^{3} + {bx}^{2} + cx + d$
We know that, a cubic equation has atleast one real root. Let real root be $α$ . Then,
$α \cdot ω \cdot ω^{2} = - \frac{d}{a} \Rightarrow α = - \frac{d}{a}$

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