# If α and β are the zeroes of a polynomial

Question:

If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x)=5 x^{2}-7 x+1$, find the value of $\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)$

Solution:

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, $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}$

$=\frac{\frac{7}{5}}{\frac{1}{5}}$

$=7$