Question

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

Answer 1

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

Roots of the required equation are cubes of the roots of given equation.

$\therefore y=x^3⟹x=\sqrt[3]{3y}$

Substituting value of $x$ in (1), we get

$(\sqrt[3]{3y}{)}^3+q(\sqrt[3]{3y})+r=0$

$⟹q\left(\sqrt[3]{3y}\right)=-(y+r)$

$⟹q^{3}y=-(y^3+3y{r}^2+3y^{2}r+r^3)\left[\text{Taking cubes on both sides}\right]$

$⟹y^3+3r{y}^2+(3r^2+q^3)y+r^2=0$