Question

If the roots of x^{​​​​​​3}+px^{​​​​​2}+qx+r=0 are in G.P find the relation between p,q,r

Answer 1

Let the roots be a/t, a and at.

Product of the roots = (a/t)(a)(at)=(a/t)(a)(at)= a3=−ra3=−r

Sum of the roots = a[1/t+1+t]=−pa[1/t+1+t]=−p => [1/t+1+t]=−p/a[1/t+1+t]=−p/a

Coefficient of x = sum of combinations of 2 roots = a(a/t)+a(at)+at(a/t)=a(a/t)+a(at)+at(a/t)= a2[1/t+t+1]=qa2[1/t+t+1]=q

=> a2(−p/a)=qa2(−p/a)=q => −ap=q−ap=q => a=−q/pa=−q/p

=>a3=−q3/p3=−ra3=−q3/p3=−r

=> q3=rp3 q3=rp3