If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x)=5 x^{2}-7 x+1$, find the value of $\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)$
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes $=\frac{-(\text { coefficient of } x)}{\text { coefficent of } x^{2}}$ and Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$
$\therefore \alpha+\beta=\frac{-(-7)}{5}$ and $\alpha \beta=\frac{1}{5}$
$\Rightarrow \alpha+\beta=\frac{7}{5}$ and $\alpha \beta=\frac{1}{5}$
Now, $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}$
$=\frac{\frac{7}{5}}{\frac{1}{5}}$
$=7$
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