Lab 1 - History of Algorithms

1. Write a Python program in Google Colab that implements the new method for solving quadratic equations

• A new way to make quadratic equations easy - MIT Technology Review
• A Different Way to Solve Quadratic Equations - youtube
• A Simple Proof of the Quadratic Formula Po-Shen Loh - arxiv

and apply the algorithm to

• x^2 – 2x +1
• x^2 – 3x +2
• x^2 +x -2
• x^2 –x -2

Upload the Google Colab link in the text box of the UN Moodle assignment 1. Google Colab link (do not share with emails, see note below)

2. Write a Python program in Google Colab that solves any fifth order polynomial equation that has positive integer roots. An apply these to x^5 – 15 x^4 + 85 x^3 - 225 x^2 + 274 x -120 and to x^5 – 14 x^4 + 72 x^3 - 166 x^2 + 167 x -60. Hints: In Python you can perform synthetic or symbolic division of polynomials https://stackoverflow.com/questions/26173058/division-of-polynomials-in-python

Upload the Google Colab link in the text box of the UN Moodle assignment as 2. Google Colab link (do not share with emails, see note below)

3. The root-finding numeric Newton method when applied to complex polynomial defines a Newton Fractal (a class of Julia set) coloring each complex plane point, depending of the number of iterations to converge or depending to what root converges, if the algorithm starts in that complex plane point . Modify the following Python program in Google Colab and input parameters to paint the following Newton fractals for f(Z) = Z^4-1

• Ask Math Anything, with Po-Shen Loh - x^3 =1 (cube roots of one) min 49:00 - youtube
• This Professor Will Teach You Math for Free While You're Social Distancing - popular mechanics

If you draw this other Newton fractals, except P(Z) = Z^3-1 colored by root reached, you get an extra credit for each one.

Upload the Google Colab link or links in the text box of the UN Moodle assignment as 3. Google Colab link or links (do not share with emails, see note below)

4. Given that there are root-finding numeric algorithms (Newton, bisection, etc) that can be applied to any continuous real or complex function, this includes polynomials equations of any degree, why these cannot be considered methods of solving polynomial equations of degree 5 rendering the Galois result false?

Type your answer in the text box of the UN Moodle assignment as as text 4 . Answer (do not share with emails, see note below)

• Ask Math Anything, with Po-Shen Loh - x^3 =1 (cube roots of one) min 49:00 - youtube
• This Professor Will Teach You Math for Free While You're Social Distancing - popular mechanics