Question

If α and β are the zeros of the quadratic polynomial f(x) = x² − 3x − 2, find a quadratic polynomial whose zeros are $\frac{1}{2\mathrm{\alpha }+\mathrm{\beta }}\mathrm{and}\frac{1}{2\mathrm{\beta }+\mathrm{\alpha }}$.

Answer 1

Since are the zeros of the quadratic polynomial

The roots are and

Let S and P denote respectively the sum and the product of zero of the required polynomial . Then,

Taking least common factor then we have ,

By substituting and we get ,

By substituting and we get ,

Hence ,the required polynomial is given by

Hence, the required equation is Where k is any non zero real number.