Algebraic Number/Examples/Cube Root of (2 plus Root 2)

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Example of Algebraic Number

$\sqrt [3] {2 + \sqrt 2}$ is an algebraic number.


Proof

Let $x = \sqrt [3] {2 + \sqrt 2}$.

We have that:

\(\ds x^3\) \(=\) \(\ds 2 + \sqrt 2\)
\(\ds \leadsto \ \ \) \(\ds x^3 - 2\) \(=\) \(\ds \sqrt 2\)
\(\ds \leadsto \ \ \) \(\ds \paren {x^3 - 2}^2\) \(=\) \(\ds 2\)
\(\ds \leadsto \ \ \) \(\ds x^6 - 4 x^3 + 4\) \(=\) \(\ds 2\)
\(\ds \leadsto \ \ \) \(\ds x^6 - 4 x^3 + 2\) \(=\) \(\ds 0\)

Thus $\sqrt [3] {2 + \sqrt 2}$ is a root of $x^6 - 4 x^3 + 2 = 0$.

Hence the result by definition of algebraic number.

$\blacksquare$


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