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Question

Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of polynomial f(x)=px2+qx+x,p0,r0

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Solution

If α,β are the zeroes of polynomial f(x)=px2+qx+r
α+β=qp and αβ=rp, then the required polynomial has the zeroes as 1α,1β
sum of its zeroes =1α+1β=α+βαβ
=qprp=qr
=ba say
product of zeroes =1α,1β=1αβ=1rp=pr=ca say
if a=1, then b=qr,c=pr in ax2+bx+c
the required polynomials is (x2+qrx+pr)

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