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If alpha and beta are zeros of the polynomial f(x)=x^-px+q, prove that
Alpha^2/beta^2+beta^2/alpha^2 = p^4/q^2 -4p^2/q +2

Answer :

kvnmurty
         f(x) = x² - p x + q
 α and β are the roots of the above equation.
  to find   α² / β² +  β² / α² = ?

   α = [ p + √(p² - 4q)  ]  / 2        and    β = [ p - √(p² - 4 q) ] / 2

so,      α + β = p      and    α β  =  q          and     α² β²  = q²

   =>    α² + β²   =  (α+β)² - 2 αβ  =  p² - 2 q

   =>    α⁴ + β⁴  =  (α² + β²)² - 2 α²β²  =  (p² - 2q)² - 2 q²
                       =  p⁴ - 4 p² q + 4 q² - 2 q²
                       =  p⁴ - 4 p² q + 2 q²
 
NOW ,  α² / β² + β² / α² =  [ α⁴  + β⁴ ] / α² β² = 
                     = [ p⁴ - 4 p² q + 2 q² ] /  q² 

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