Question

If α and β are the roots of the polynomial f(x) = x² – 2x + 3, find the polynomial whose roots are `(alpha - 1)/(alpha + 1), (beta - 1)/(beta + 1)`

Answer 1

Sum of the roots

= `(alpha - 1)/(alpha + 1) + (beta - 1)/(beta + 1)`

= `((alpha - 1)(beta + 1) + (beta - 1)(alpha + 1))/((alpha + 1)(beta + 1))`

= `(alphabeta + alpha - beta - 1 + alphabeta + beta - alpha - 1)/(alphabeta + alpha + beta + 1)`

= `(2alphabeta - 2)/(alphabeta + alpha+ beta + 1)`

= `(2(3) - 2)/(3 + 2 + 1)`

= `4/6`

= `2/3`

Product of the roots

= `(alpha - 1)/(alpha + 1) xx (beta - 1)/(beta + 1)`

= `((alpha - 1)(beta - 1))/((alpha + 1)(beta + 1))`

= `(alphabeta - alpha - beta + 1)/(alphabeta + alpha + beta + 1)`

= `(alphabeta - (alpha + beta) + 1)/(alphabeta + (alpha + beta) + 1)`

= `(3 - 2 + 1)/(3 + 2 + 1)`

= `2/6`

= `1/3`

The quadratic polynomial is

x² – (sum of the roots) x + products of the roots = 0

`x^2 – (2/3) x + 1/3` = 0

3x² – 2x + 1 = 0