If alpha and beta are the zeroes of the quadratic polynomial f(x) = x2 - px q, prove that alpha2/beta2 beta2/alpha2 = p4/q2 - 4p2/q 2.
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The given quadratic polynomial f(x) = x2 − px qComparing this quadratic polynomial with standard quadratic polynomial ax2 bx c, we get,a = 1 ; b = −p ; c = qSince α & β are the zeroes of the given polynomialthen, α β = −ba = −(−p)1 = pand αβ = ca = q1 = qNow, α2β2 β2α2= α4 β4α2 β2= (α2 β2)2 − 2 α2 β2α2 β2= [(α β)2 − 2 αβ]2 − 2 (αβ)2(αβ)2= [p2 − 2q]2 − 2q2q2= p4 4q2 − 4p2q − 2q2q2= p4 − 4p2q 2q2q2= p4q2 − 4p2qq2 2q2q2= p4q2 − 4p2q 2
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The given quadratic polynomial f(x) = x2 − px qComparing this quadratic polynomial with standard quadratic polynomial ax2 bx c, we get,a = 1 ; b = −p ; c = qSince α & β are the zeroes of the given polynomialthen, α β = −ba = −(−p)1 = pand αβ = ca = q1 = qNow, α2β2 β2α2= α4 β4α2 β2= (α2 β2)2 − 2 α2 β2α2 β2= [(α β)2 − 2 αβ]2 − 2 (αβ)2(αβ)2= [p2 − 2q]2 − 2q2q2= p4 4q2 − 4p2q − 2q2q2= p4 − 4p2q 2q2q2= p4q2 − 4p2qq2 2q2q2= p4q2 − 4p2q 2
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