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Question:

True/False?

The polynomial $x^3+3x-2\pi$ is irreducible over $\mathbb{R}$

Hint:

When is a odd degree polynomial irreducible over $\mathbb{R}$?

Discussion:

A polynomial $p(x)$ is irreducible over a field if it can not be written as product of two non-trivial polynomials having coefficients from the same field.

In other words, a polynomial $p(x) \in \mathbb{F}[x]$ of degree $k$ is irreducible over $\mathbb{F}$ if there is no non-constant polynomial of degree $<k$ dividing $p(x)$ in $\mathbb{F}[x]$.

For a polynomial $p(x)$ of odd degree $\ge 3$ over $\mathbb{R}$ there is always a real root $\alpha$. This is because the complex roots always occur in conjugate pairs. Therefore, $(x-\alpha )$ divides $p(x)$.

Therefore, $p(x)$ is not irreducible.