Real roots of a seven degree polynomial with integer coefficients

Question

What are the all possible no. of real roots of a seven degree polynomial with integer coefficients ?

I was reading a mathematical physics book which mentioned that the possible no. of real roots of a seven degree polynomial with integrer coefficients are 1, 3, 5, 7. I do not understand why ?

Answer 1

Let $p$ be an odd degree polynomial, then using the intermediate value theorem, $p$ has at least one root in $\mathbb{R}$. Furthermore, notice that for a real polynomial, strictly complex roots comes in pairs, since $p$ is invariant by the complex conjugation. Hence, the number of possible roots counted with multiplicity for $p$ are the odd integers between $0$ and $\deg(p)$.

Answer 2

This is incorrect. Consider the polynomial $x^7-x^6$, which has two real roots, namely 0 (multiplicity 6) and 1 (multiplicity 1).

Answer 3

Most of the time the number of real roots must be odd. To see why, imagine the graph. We can assume the coefficient of the leading term is positive. Then as $x$ grows, the graph is eventually above the $x$ axis and stays there. When $x$ is way off to the left the graph is below the axis.

Since it starts below and ends above it must cross an odd number of times.

That's most of the time. But the graph might be tangent to the $x$ axis, as in the comment from @HansEngler. To get the count right you have to count that kind of root with its proper multiplicity. That will be even if the root is a local maximum or minumum and odd if the root is an inflection point.