Question

Find the equation whose roots are the cubes of the roots of \(x^3 + 3x^2 + 2 =0\)

Answer 1

Replace \(x\) by \(x^{\frac{1}{3}}\) then given equation becomes \(3x^{\frac{2}{3}} =-(x+2)\)
On cubing both sides, we get \(27x^2 =(-(x+2))^3\)
\(\Rightarrow 27x^2 =-(x^3 +6x^2 +12x + 8)\)
\(\Rightarrow x^3 + 33x^2 + 12x + 8 =0\)
\(\therefore\) The required equation is \(x^3 + 33x^2 + 12x + 8 =0\)