If 1, 2, 3 are the roots of ${\mathrm{ax}}^3 + {\mathrm{bx}}^2 + \mathrm{cx} + \mathrm{d} = 0$ then the roots of ${\mathrm{ax}}^3 + 2{\mathrm{bx}}^2 + 4\mathrm{cx} + 8\mathrm{d} = 0$ are

$\begin{array}{l}Letf\left(x\right)=a{x}^3+b{x}^2+cx+d=0\\ \mathrm{Re}placexby\frac{x}{2}then\\ f\left(\frac{x}{2}\right)=a\left(\frac{x^3}{8}\right)+b\left(\frac{x^2}{4}\right)+c\left(\frac{x}{2}\right)+d=0\\ \Rightarrow a{x}^3+2b{x}^2+4cx+8d=0\\ \therefore Therootsoff\left(\frac{x}{2}\right)=2timestherootsoff\left(x\right)\\ =2,4,6\end{array}$