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

(i) Since and are the zeros of the quadratic polynomial

Product of the zeros =

Let S and P denote respectively the sums and product of the polynomial whose zeros

Therefore the required polynomial f (x) is given by

Hence, the required equation is.

(ii) Since and are the zeros of the quadratic polynomial

Product of the zeros =

Let S and P denote respectively the sums and product of the polynomial whose zeros

By substituting and we get ,

The required polynomial f (x) is given by,

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