Question

If $x=3+3^{\frac{1}{2}}+3^{\frac{2}{3}}$. Find $x^3-9x^2+18x-12$.

Answer 1

We have, $x=3+3^{1/2}+3^{2/3}$

$(x-3)=3^{1/2}+3^{2/3}$

$(x-3{)}^3=(3^{1/2}+3^{2/3}{)}^3$

$\Rightarrow x^3-27-9x^2+27x=3+9+3\times 3(3^{1/2}+3^{2/3})$

$\Rightarrow x^3-27-9x^2+27x=12+9(x-3)$

$\Rightarrow x^3-27-9x^2+27x-12-9x+27=0$

$\Rightarrow x^3-9x^2+18x-12=0$