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Decide whether the polynomial

x3 +23x2 + 3x −  5


is irreducible  over Q or not

 Sep 12, 2018

Apply the rational root theorem


\(\text{let }p(x) = a_0 + a_1 x + \dots + a_n x^n\)


\(\text{then if }x = \dfrac{p}{q} \text{ written in lowest terms is a root of p(x), }\\ \text{it must be that p divides }a_0 \text{and that q divides }a_n\)


\(\text{here we have }a_0 = -5,~a_3=1\)


\(\text{the only possible p's are }\pm 1,~\pm 5\\ \text{and the only possible q's are }\pm 1\)


\(\text{thus the only possible rational roots are }\pm 1,~\pm 5 \\ \text{and we just have to try them all to see if any are actually roots.}\)


\(p(-1)=14, ~p(1) = 22,~p(5)=710,~p(-5)=430 \\ \text{i.e. none of these are in fact roots and thus p(x) is irreducible over the rational numbers}\)

 Sep 12, 2018

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