If $α$ and $β$ are the zeros of the quadratic polynomial $f(x)\ =\ ax^2\ +\ bx\ +\ c$, then evaluate:
$α^2β\ +\ αβ^2$
Given:
$α$ and $β$ are the zeros of the quadratic polynomial $f(x)\ =\ ax^2\ +\ bx\ +\ c$.
To do:
We have to find the value of $α^2β\ +\ αβ^2$.
Solution:
The given quadratic equation is $ax^2+bx+c=0$, where a, b and c are constants and $a≠0$.
Sum of the roots $= α+β = \frac{-b}{a}$.
Product of the roots $= αβ = \frac{c}{a}$.
We know that,
$α^2β+αβ^2=αβ(α+β)$
$=\frac{c}{a}(\frac{-b}{a})$
$=-\frac{bc}{a^2}$
The value of $α^2β\ +\ αβ^2$ is $-\frac{bc}{a^2}$.
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