Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?
The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$
(or over $\mathbb{Z}$ as Gauss' lemma can be used.) and have integer coefficients.
I have no idea how to generate such a polynomial.
I've tried things along the lines of
$f(x) = x^5(x^2 - 5) + 5$
As these are easily irreducible with Eisenstein Criterion $(p = 5)$
Peter has kindly shown (via brute force) that $$x^7+x^6-3x^5-x^4-2x^3-3x^2+x+1$$
Matches said criterion, yet has yet to show that this is irreducible. Can anyone do this?
Thank you to achille hui who has given me my answer of:
$f(x) = x^7+1000003(x−1)(x−2)(x−3)(x−4)(x−5)$
which is irreducible by Eisenstein's with $p = 1000003$.
Answer
Another possibility is taking any monic integer polynomial $m(x)$ with 5 real distinct roots (different from zero) and consider polynomial of the form $x^7+pm(x)$ for sufficiently large prime p. e.g.
$f(x) = x^7+1000003(x−1)(x−2)(x−3)(x−4)(x−5)$
which is irreducible by Eisenstein's with $p = 1000003$.
Credit - achille hui. Posted as anwser rather than comment for clarity.
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