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

Given:

$α$  and  $β$  are the zeros of the quadratic polynomial $f(x)\ =\ x^2\ -\ 1$.

To do:

We have to find the quadratic polynomial having $\frac{2α}{β}$ and $\frac{2β}{α}$ as its zeros.  

Solution:

We know that, 

The standard form of a quadratic equation is $ax^2+bx+c=0$, where a, b and c are

constants and $a≠0$

Comparing the given equation with the standard form of a quadratic equation, 

$a=1$, $b=0$ and $c=-1$

Sum of the roots $= α+β = \frac{-b}{a} = \frac{-0}{1}=0$.

Product of the roots $= αβ = \frac{c}{a} = \frac{-1}{1}=-1$.

Let the sum and product of the zeros of the given quadratic equation be $S$ and $P$.

Therefore,

$S= \frac{2α}{β}+\frac{2β}{α}$

$=\frac{2α^2+2β^2}{αβ}$

$=\frac{2(α^2+β^2)}{αβ}$

$=\frac{2((α+β)^2-2αβ)}{αβ}$

$=\frac{2((0)^2-2(-1))}{-1}$

$=\frac{4}{-1}$

$=-4$

$P=\frac{2α}{β}\times \frac{2β}{α}$

$=4$

The quadratic polynomial having the sum of the roots $S$ and product of the roots $P$ is $f(x)=k(x^2-(S)x+P)$, where $k$ is any non-zero real number.

Therefore,

$f(x)=k(x^2-(-4)x+4)$

$f(x)=k(x^2+4x+4)$

The required quadratic polynomial is $f(x)=k(x^2+4x+4)$, where $k$ is any non-zero real number.

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Updated on: 10-Oct-2022

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