Polynomials and number theory (I guess...)

[Xiphias]

This is a more complicated problem: Is there a formula (algorithm) to create polynomials with both whole nunbered roots and whole numbered local max/min points?


In math notation: How to make a P(x) such that if P(a) = 0 then a \in Z AND if dP/dx P(b)=0 then b \in Z ?

- Xiphias.

 

[Bob Brown MSEE]

You need factors like these, that also factors (over the integers) after derivative is calculated...

Example: Here is one ...
(x-1)(x-1)(x-3)(x-3)

I don't see a simple way to do that in general.

 

[Bob Brown MSEE]

You need factors like these, that also factors (over the integers) after derivative is calculated...

Example: Here is one ...
(x-1)(x-1)(x-3)(x-3)

I don't see a simple way to do that in general.

Here's a generalization of this one way...
(xx-mx)^n, m and n integers with m even parity.

 

[Xiphias]

Nice! (x^2-xm)^n would never get you a third degree polynomial though... Is it even possible for third degree polynomial of all three roots should be distinct?

 

[soroban]

Hello, Xiphias!

I don't know of any general formula.
I just derived one for quadratic functions.

Is there a formula (algorithm) to create polynomials
with both integer roots and integer local max/min points?

A quadratic function with integral roots could have the form:

. . \(\displaystyle f(x) \:=\: (x-a)(x-b) \:=\: x^2 - (a+b)x + ab\)

I intentionally disregarded a leading coefficient.


Its derivative is: .\(\displaystyle f'(x) \:=\:2x - (a+b)\)

It has a minimum at: .\(\displaystyle x \:=\:\dfrac{a+b}{2}\)
The vertex is always halfway between the x-intercepts.

For an integral minimum point, \(\displaystyle a\) and \(\displaystyle b\) must have the same parity
. . (both odd or both even).


Similar work for a cubic function became unwieldy.
You can try it if you dare.