Possible integer values for coefficients of cubic equation with given root

[anemone]

Given $a,\,b,\,c$ and $d$ are all integers such that $x=\sqrt[3]{\sqrt{8}+4}-\sqrt[3]{\sqrt{8}-4}$ is a root to the equation $ax^3+bx^2+cx+d=0$. Find the possible values for $(a,\,b,\,c,\,d)$.

 

[kaliprasad]

Given $a,\,b,\,c$ and $d$ are all integers such that $x=\sqrt[3]{\sqrt{8}+4}-\sqrt[3]{\sqrt{8}-4}$ is a root to the equation $ax^3+bx^2+cx+d=0$. Find the possible values for $(a,\,b,\,c,\,d)$.

Spoiler

$x=\sqrt[3]{\sqrt{8}+4}-\sqrt[3]{\sqrt{8}-4}$
cube both sides to get
$x^3 = \sqrt{8}+4 - (\sqrt{8}- 4) - 3\sqrt[3]((\sqrt{8}+4)(\sqrt{8}-4))x$
or $x^3=8-3\sqrt[3](8-16)x=8+6x$
or $x^3- 6x -8=0$ so $a = t, b = 0, c = -6t , d= -8t $ where t is any non zero integer

 

[Theia]

Spoiler

$$\begin{align*}x^3 &= \sqrt{8} + 4 - \sqrt{8} + 4 + 3\left( -\left(\sqrt[3]{\sqrt{8} + 4}\right)^2 \sqrt[3]{\sqrt{8} - 4} + \sqrt[3]{\sqrt{8} + 4} \left(\sqrt[3]{\sqrt{8} - 4}\right)^2 \right) \\
&= 8 + 3\sqrt[3]{\sqrt{8} + 4} \sqrt[3]{\sqrt{8} - 4}\left( -x \right) \\
&= 8 - 3\sqrt[3]{8-16}x \\
&= 8 + 6x \end{align*}$$

Thus

$$a = t,\ b = 0,\ c = -6t,\ d = -8t,\ t \in \mathbb{R} \smallsetminus 0.$$

 

[anemone]

Hi kaliprasad and Theia!

Very well done to the both of you! And thanks for participating!