Question

Find the polynomial whose zeros are $\alpha =1/3$ and $\beta =1/2$

Answer 1

Given zeroes of the polynomial$=\alpha =\frac{1}{3}$ and $\beta =\frac{1}{2}$

Sum of the roots$=\alpha +\beta =\frac{1}{3}+\frac{1}{2}=\frac{2+3}{6}=\frac{5}{6}$

Product of the roots$=\alpha \beta =\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$

The polynomial is $x^2-(\alpha +\beta )x+\alpha \beta =0$

$x^2-\frac{5x}{6}+\frac{1}{6}$

or $6x^2-30x+1$ is the required polynomial.