Just one more small geometry task to get to know Maple a little better:
In the 1st quadrant, point A is on the y-axis and points B, D, C are at a positively increasing distance from the origin on the x-axis. Let BD = DC = s/2 and angle ADB = 45°. Point D is therefore the center of BC. The area of ​​triangle ABC is 60, line AD = x and the length of AC = 19. We are looking for the length of line AB = y.

A student wakes up at the end of the lecture and just catches the professor saying:
"... and the roots form an arithmetic sequence."

On the board there is a 5th degree polynomial as homework, but unfortunately the student only manages to write down
x^5 - 5x^4 - 35x^3 +
before the professor wipes the board.

But the student still finds all the roots of the polynomial.

And the roots now have to be calculated.

If you draw a chord in any curve, when the latter becomes infinitely small, the ratio of the surface segment to the triangle formed by the chord and the associated tangents is 2:3.

(Source: Archive of Mathematics and Physics, editor Johann August Grunert, 31st part of 1858, pp. 449-453, "On a remarkable general theorem on curves",
Author: Andreas Völler)

(The curve may be assumed to be sufficiently differentiable.)

With the attached task I would like to learn how Maple handles polynomials and plots. For this I have chosen a task with an interesting history from 1593. Adriaan van Roomen posed it to F. Vieta, who solved it after a short consideration.

My questions are:
1.) How are very long terms entered and displayed in Maple in an appropriate manner?
2.) How are the graphs of several functions displayed in the same coordinate system?
3.) Which graphics settings must be selected for differentiable functions in order to obtain nice, rounded curves rather than angular ones from the numerical result?
4.) Can the apparently random oscillations of the curve at the end of the interval be suppressed?
5.) The curves for p(x) and sinnx(x) are theoretically identical, since p(x) is the trigonometric expansion of sinnx(x). Their graphs must therefore be identical. How can this be displayed?
6.) The polynomial has many real zeros. How can the zeros be clearly presented in a table?

AF_20240909.mw

As a newbie in Maple2024, my next step is to try to solve a Diophantine equation. Attempts with isolve failed. It is the famous task B3 from the IMO of 1988:

If for integers x and y the fraction (x^2+y^2)/(1+x*y) is a positive integer, then it is actually a square number.

The solution is well known. But I would like to learn how to use Maple using this example and would like some advice.

Best regards, Alfred_F