First slide

Theory of expressions

Question

If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ , then the value of $α^{3} + β^{3}$ is

Moderate

A

$\frac{3 a b c + b^{3}}{a^{3}}$
B

$\frac{a^{3} + b^{3}}{3 a b c}$
C

$\frac{3 a b c - b^{3}}{a^{3}}$
D

$\frac{- (3 a b c + b^{3})}{a^{3}}$

Solution

We have
$α + β = \frac{- b}{a}, α β = \frac{c}{a}$
Now, $α^{3} + β^{3} = (α + β)^{3} - 3 α β (α + β)$
$= {(\frac{- b}{a})}^{3} - \frac{3 c}{a} (\frac{- b}{a}) = \frac{3 a b c - b^{3}}{a^{3}}$

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