Question

If α and β are the zeroes of a polynomial f(x) = 5x² − 7x +1, find the value of $\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)$

Answer 1

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = $\frac{-\left(\mathrm{coefficient}\mathrm{of}\mathit{}x\right)}{\mathrm{coefficent}\mathrm{of}{x}^2}$ and Product of zeroes = $\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficent}\mathrm{of}{x}^2}$
$$\begin{gathered} \therefore \alpha +\beta =\frac{-\left(-7\right)}{5}\mathrm{and}\alpha \beta =\frac{1}{5} \\ \Rightarrow \alpha +\beta =\frac{7}{5}\mathrm{and}\alpha \beta =\frac{1}{5} \\ \mathrm{Now},\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta } \end{gathered}$$

$$\begin{gathered} =\frac{{\displaystyle \frac{7}{5}}}{{\displaystyle \frac{1}{5}}} \\ =7 \end{gathered}$$