Solving Third Degree Polynomial Equations Michel Beaudin michel

Solving Third Degree Polynomial Equations Michel Beaudin, michel. beaudin@etsmtl. ca Frédérick Henri, frederick. henri@etsmtl. ca École de technologie supérieure, Montréal, Québec, Canada TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 1

Abstract Starting with some concrete examples, we use TI-Nspire CX CAS to solve third degree polynomial equations with real coefficients. Then we compare the answers obtained to the ones provided by other CAS. Using Cardano’s method or François Viète’s formulas, we create a homemade function allowing Nspire to produce clear and compact answers for the solutions of a third degree polynomial equation: "Everything should be made as simple as possible, but not simpler". TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 2

Table of contents 1. Two examples showing CAS imperfections 2. An algorithm to beautify the solutions 3. Solving a third degree polynomial equation with 1 real root 4. Solving a third degree polynomial equation with 3 real roots 5. Implementation of a new TI-Nspire function 6. Conclusion TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 3

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 4

Two examples showing CAS imperfections – first example TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 5

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 6

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 7

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 8

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 9

Two examples showing CAS imperfections – first example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 10

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 11

Two examples showing CAS imperfections – second example TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 12

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 13

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 14

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 15

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 16

Two examples showing CAS imperfections – second example § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 17

An algorithm to beautify the solutions § TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 18

An algorithm to beautify the solutions Choose a 3 rd degree function Get rid of the 2 nd degree term 1 real root Ascertain the number of real roots 3 real roots Transform into a 6 th degree polynomial equation Apply a trigonometric substitution Transform into a 2 nd degree polynomial equation Solve the equation Select one solution of the previous equation Compute the roots of the original function Compute the 3 cubic roots of this solution Compute the roots of the original function TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 19

Solving a third degree polynomial equation with 1 real root Let’s use our algorithm to solve a third degree polynomial equation that possesses a single real root. To solve the equation, we will need to accomplish 8 steps, as seen in the algorithm’s organigram. START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 20

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 21

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 22

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 23

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 24

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 25

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 26

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 27

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 28

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 29

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 30

Solving a third degree polynomial equation with 1 real root § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 8 7 2021 -06 -12 31

Solving a third degree polynomial equation with 3 real roots Let’s use our algorithm to solve a third degree polynomial equation that possesses 3 real roots. To solve the equation, we will need to accomplish only 6 steps, as seen in the algorithm’s organigram. START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 32

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 33

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 34

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 35

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 36

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 37

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 38

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 39

Solving a third degree polynomial equation with 3 real roots § START 2 3 TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 4 5 6 2021 -06 -12 40

Implementation of a new TI-Nspire function This new function has been implemented by Michel Beaudin and can be downloaded. Go to https: //cours. etsmtl. ca/seg/mbeaudin/ And save the TI-Nspire CX CAS library “Kit_ETS_MB” into My. Lib. The name of the function is “compact_cubic”. Live examples: on the TI-Nspire CX CAS file. TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 41

Conclusion Teaching mathematics using CAS is, as far as we are concerned, an excellent opportunity for mathematics teachers to stay enthusiastic even though the curriculum has been quite the same for the several past years! TIME 2016, UNAM, Mexico City, Mexico, June 29 th - July 2 nd 2021 -06 -12 42