First slide

Theory of equations

Question

If the sum of two roots of the equation $x^{3} - p x^{2} + q x - r = 0$ is zero, then

Moderate

A

$p q = r$
B

$q r = p$
C

$p r = q$
D

$pqr = 1$

Solution

Let the roots of the given equation be $α, β, γ$ such that
$α + β = 0$ Then,
$α + β + γ = - \frac{(- p)}{1} \Rightarrow α + β + γ = p \Rightarrow γ = p [∵ α + β = 0]$
Hence, the real root of $a x^{3} + b x^{2} + c x + d = 0$ is $\frac{d}{a}$

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