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Question

If α&β are zeros of the polynomial f(x)=x2+px+q,then find a polynomial having 1/α & 1/β as its zeros.

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Solution

f(x)=x²+px+q
Sum of roots, α+ β = -p
Product of rootsr, αβ = q
(1/α + 1/β) = (α + β) / αβ = - p / q
1/αβ = 1 / q.
If 1/α, 1/β are zeros of the quadratic polynomial then the equation is
x² -(1 / α + 1 / β)x + 1 / αβ = 0
then
x² -(-p / q)x + 1 / q = 0
i.e qx² + px + 1 = 0

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