Find the quadratic polynomials which have (α +β) and (α −β) as their zeroes.

A polynomial with zeroes a and b is of the form $\mathrm{k}\left[{\mathrm{x}}^2-(\mathrm{a}+\mathrm{b})\mathrm{x}+\mathrm{ab}\right]$, where k is a real number.

Here, sum of zeroes = (α +β) + (α −β) = 2α

Multiplication of zeroes = (α +β) (α −β) = α² − β²

Therefore, the required polynomial is k [x² − (2α)x + (α² − β²)].

For k = 1, the required polynomial is x² − (2α)x + (α² − β²).