If alpha and beta are the zeroes of the quadratic polynomial f(x) = x2 - px + q, prove that - Math - Polynomials

Question

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

Manbar Singh · Accepted Answer

The given quadratic polynomial fx = 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 = --p1 =
pand αβ = ca = q1 = qNow, α2β2 +
β2α2= α4 + β4α2 β2= α2 +
β22 - 2 α2 β2α2 β2= α + β2 - 2
αβ2 - 2 αβ2αβ2= p2 - 2q2 - 2q2q2= p4
+ 4q2 - 4p2q - 2q2q2= p4 - 4p2q + 2q2q2= p4q2 - 4p2qq2 + 2q2q2=
p4q2 - 4p2q + 2

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.

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.