Question

find the cubic equation whose roots are the cube of the roots of x^3+ax^2+bx+c=0,a,b,c€R

Answer 1

let P , Q , r are the roots of
x³ + ax² + bx + c =0 a, b , c € R

then,
P + Q + r = -a
PQ + Qr + rP = b
PQr = -c
.now,
according to question,
a unknown equation, which roots are cube of the roots of given equation .
e.g P³ , Q³ , r³ are the roots of unknown equation .
now,
equation,
x³ -( sum of roots )x² + ( sum of product of two roots )x - products of roots =0

sum of roots = P³ + Q³ + r³
= (P+Q +r){ P² +Q² +r² -(PQ +Qr +rP) -3PQr =(-a){ a² -3b} +3c
= -a³ +3ab +3c

(PQ)³ + (Qr)³ + (rP)³ = (PQ +Qr +rP){ (PQ +Qr +rP)² -3( PQ²r +r²PQ +P²Qr)} -3(PQr)³

=(b){b² -3(-c)(-a) } +3c³
=b³ -3abc +3c³

(PQr)³ = -c³

now, put this

x³ -( -a³ +3ab + 3c)x² +(b³-3abc + 3c³)x + c³