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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)`
Solution
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
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