Number of roots of special polynomials in interval $(0,1)$

Question

I proved that some special class of polynomials has only one root in interval (0,1)

For example: $x^{20}+x^{10}+3x^5-30x^{4}+2x+3=0$

Or

$2x^7-10x^5+2x^3+3x+2=0$

Or

$8x^6-2x^5-2x^4-2x^2-1=0$

My question: Is this an interesting result? How can I check for which class of polynomials we know such property?

Answer 1

Note that any polynomial (with real coefficients) which has a (real) root can be scaled and translated so that it has just one root in any given interval [may be a multiple root if the only roots of the original polynomial are multiple roots]. So there has to be some additional feature to make the observation interesting mathematically.

It would be interesting to know what is special about the class you have identified. [It won't form a linear space - the sum of two such polynomials need not have the same property]

Answer 2

Maybe if you say what special kind of polynomials could be an interesting topic ... but my first impression is that the topic looks promising

Answer 3

For a continuous function (like a polynomial) to have more than one root in [0,1], it must have a point where $f'(x)=0$ in [0,1]. This is supported by Rolle's Theorem. Any function whose derivative has no zeroes in [0,1], will also have no more than 1 zero in the interval. This will not find all types of functions that satisfy your condition, but it can help.

Polynomials have at most as many roots as their degree (other than the 0 polynomial). All polynomials can be factored in the form $(x-r_1)(x-r_2)\times\ldots\times (x-(a_1+b_1i))(x-(a_1-b_1i))(x-(a_2+b_2i))(x-(a_2-b_2i))\ldots$; i.e. you can factor an $n$th degree polynomial into $n$ binomials; some of which show their real roots, $r_i$, and others show their complex roots.

Is it interesting? I don't particularly think so, but there is some exploration that can be done.

You can think about modifying a given function to satisfy the criteria. You can also take any continuous function (it won't have vertical asymptotes in the interval) with more than 1 root in the interval and subtract it's minimum value on then interval, then it will have exactly 1 root.