Question

If $a,b,c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are $a(b+c),b(c+a),c(a+b)$.

Answer 1

Given equation, $x^3+qx+r=0\dots \dots \dots \dots \dots \dots \left(1\right)$

$a(b+c)=a(a+b+c-a)=-a^2[\because \text{sum of the roots is 0}]$

$⟹y=-x^2\text{or}{x}^2+y=0\dots \dots \dots \dots ..\left(2\right)$

Multiplying (2) by $x$ and subtracting from (1), we get

$(q-y)x+r=0⟹x=\frac{r}{y-q}$

Substituting value of $x$ in (2), we get

$y^3-2q{y}^2+q^{2}y+r^2=0$