If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = a x^{2} + b x + c$ , then evaluate: $α^{2} β + α β^{2}$

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Solution

If $α$ and $β$ are zeros of $f (x) = a x^{2} + b x + c$ , then evaluate
$a^{2} β + α β^{2}$
Observe that we have $α + β = \frac{- b}{a}$ and $α β = \frac{c}{a}$
Thus
$α^{2} β + α β^{2} = α β (α + β) = \frac{c}{a} (\frac{- b}{a}) = \frac{- b c}{a^{2}}$

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