If α and β are the zeroes of a polynomial f(x)


If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x)=6 x^{2}+x-2$, find the value of $\left(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right)$



By using the relationship between the zeroes of the quadratic ploynomial.
We have,

Sum of zeroes $=\frac{-(\text { coefficient } \text { of } x)}{\text { coefficent } \text { of } x^{2}}$ and Product of zeroes $=\frac{\text { constant term }}{\text { coefficent of } x^{2}}$

$\therefore \alpha+\beta=\frac{-1}{6}$ and $\alpha \beta=-\frac{1}{3}$

Now, $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}$

$=\frac{\alpha^{2}+\beta^{2}+2 \alpha \beta-2 \alpha \beta}{\alpha \beta}$

$=\frac{(\alpha+\beta)^{2}-2 \alpha \beta}{\alpha \beta}$





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