<s>[INST] write the following as a polynomial in $\alpha$ with rational coefficient Let $\alpha = \sqrt[3]{2}$ 
write $\frac{\alpha^2 -1}{\alpha + 2}$ as a polynomial in $\alpha$ with rational coefficient.
First I write the following:
$\frac{\alpha^2 -1}{\alpha + 2} = \frac{2^{2/3} - 1}{2^{1/3} + 2}$
Then I proceed to manipulate the R.H.S hoping to achieve all rational terms. But it's been a mess so far. Have interpret the question wrongly or is there some algebra trick here I need to do.
Any help or insight is deeply appreciated.
 [/INST] We have 
$$
\alpha^3-2=0 \implies \alpha^2+8=10 \implies \frac{1}{\alpha+2}=\frac{\alpha^2-2\alpha+4}{10}
$$
Thus 
$$
\frac{\alpha^2 -1}{\alpha + 2}=\frac{(\alpha^2-1)(\alpha^2-2\alpha+4)}{10}=\frac {1}{10}\,{\alpha}^{4}-\frac{1}{5}\,{\alpha}^{3}+\frac{3}{10}\,{\alpha}^{2}+\frac{1}{ 5}\,\alpha-\frac{2}{5}
$$
</s>