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Question

If a,b,c are the roots of x3+qx+r=0, form the equation whose roots are bc+cb,ca+ac,ab+ba.

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Solution

Given equation, x3+qx+r=0(1)

bc+cb=b2+c2bc=aa2+b2+c2a2abc=a(a)22aba2abc=a(a2+2qr)

y=x(x2+2qr)

x3+2qxry=0..(2)

Subtracting (1) from (2), we have

qxryr=0x=(rq)(1+y)..(3)

Substituting value of x from (3) in (1), we get

(rq)3(1+y)3+r(qq)(1+y)+q3=0

r2y3+3r2y2+(2r2+q3)y+(r2+2q3)=0


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