Not sure where to go since there's a fraction 1/2. Could I look at $8x^3 - 6x -1$?
2026-03-30 01:51:41.1774835501
Is the polynomial $4x^3 - 3x - 1/2 $ irreducible over $\mathbb{Q}$?
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Since $8x^3-6x-1=(2x)^3-3(2x)-1$, your polynomial is irreducible if and only if the polynomial $x^3-3x-1$ is irreducible. By the rational root theorem, the only rationals that can possibly be roots of this polynomial are $\pm1$. But none of them is. And a cubic polynomial over $\mathbb Q$ without rational roots is irreducible.