If $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x)=x^2−px+q$, then write the polynomial having $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ as its zeroes.
Given: $\alpha$ and $\beta$ are the zeroes of the polynomial $f(x)=x^2−px+q$.
To do: To write the polynomial having $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ as its zeroes.
Solution:
The equation having the roots as $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is $f(\frac{1}{x})$.
​
$\Rightarrow f(\frac{1}{x})=(\frac{1}{x})^{2}-px+q $
$\Rightarrow f(\frac{1}{x})=qx^2−px+1$.
Therefore the equation having $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ as roots is $qx^2−px+1$.
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