Let's discuss a problem based on Checking irreducibility over \(\mathbb{R}\) from TIFR 2013, Problem 37. Try this Problem and then read the solution.
Question: Checking irreducibility over \(\mathbb{R}\)
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.
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