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 ?

Because the possible numbers of non-real roots are $0$, $2$, $4$, $6$. — Angina Seng, Apr 28 '18 at 13:13
I assume you mean to add that the coefficients of the polynomial are real. Given that, then the complex roots come in pairs. — lulu, Apr 28 '18 at 13:13
And you mean to add (per Hans Engler) that the roots are counted with multiplicity. — John Hughes, Apr 28 '18 at 13:38

score 4 · Accepted Answer · answered Apr 28 '18 at 13:14

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 $\deg(p)$.

score 1 · Answer 2 · answered Apr 28 '18 at 13:16

1

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

answered Apr 28 '18 at 13:16

Hans Engler

15,439

score 1 · Answer 3 · answered Apr 28 '18 at 13:32

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.

Real roots of a seven degree polynomial with integer coefficients

3 Answers3