If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = x^{2} - 1$ , find a quadratic polynomial whose zeros are $\frac{2 α}{β}$ and $\frac{2 β}{α}$ .

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Solution

We have, $f (x) = x^{2} - 1 = 0$

$\Rightarrow$ $(x - 1) (x + 1) = 0$

$\Rightarrow$ $x = 1$ or $- 1$

$\Rightarrow$ $α = - 1$ , $β = 1$

New Roots are $\frac{2 α}{β} a n d \frac{2 β}{α}$
$= - 2$ or $- 2$

$\Rightarrow$ Sum of new roots $= - 4$

$\Rightarrow$ Product of new roots $= 4$

$∴ f (x) = (x^{2} + 4 x + 4)$

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