
                       ******************************
                       * POLFRAC.83G for the TI-83  *
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ACKNOWLEDGMENT

POLFRAC.83G is a group file containing several programs written by Philippe
Fortin of Nice, France and is released to the public domain.  You may copy
and change these programs.                                                      .

INTRODUCTION

This group of programs covers all the basic needs for polynomial and rational
functions study :
 - expanded form
 - factoring
 - derivative function
 - roots
 - sign study
 - variations array
 - simplification
 - integration (polynomial function)
 - asymptote (rational function)
 
All these features are available from one single program, thanks to the
'menu' TI-83 instruction.
 

HOW TO USE POLFRAC

The main program is AASTART
 
All the other programs included in POLFRAC.83G are subprograms of AASTART:
AAINI   initialize vars
DEMO    demo program (you may use it alone,
        or using the DEMO option of AASTART)
POLxxxx progs are for P(x) study
FRxxxx  progs are for P(x)/Q(x) study
SPxxxx  progs are for displays of polynomials function, conversion to frac,
        etc.
PTxxxx  progs are used to display sign and variation arrays.
 
1) DEMO
 
First you may have a quick look at what may be done using the DEMO
option of main program : You just need to run AASTART and choose 3:DEMO
to do that.

First you will get a study of P(x)=x^3-3x factorization and then
a study of derivative function sign, and then array of variations.
 
PAUSES IN PROGRAM
 
 - Press ENTER each time you get a >>> display to go forward.
 
 - When an array is displayed (variation array or sign array) you may
 use cursor keys to move cursor on an interesting position of array to
 get a more accurate value (variations array) or the expression of a
 factor (sign array). When you have got all the info needed, just press
 ENTER key to continue.
 
 
2) POLYNOMIAL FUNCTIONS : run AASTART and select 1:P(X) option.
 
This program mainly uses 3 polynomial functions P1, P2, P3.
You may define P1 & P2 using DEF POLY option. You may also store results of
polynomial computation (derivative, integral) in P1, P2 or P3.
 
To define P1 or P2, there are two possibilities :
 
- Input of degree and coef. using DEF P1 or DEF P2 options of DEF POLY menu
 
- Expanding the polynomial expressions, stored in Y1 or Y2
  BEFORE running AASTART, using Y1 -> P1 or Y2 -> P2 of DEF POLY menu.
 
Let's look at two examples :
 
EX. 1 : You want to get the derivative function of x^3-3x and then factor
this derivative function.
 
Run AASTART, select P(X), then select DEF POLY, and DEF P1. Enter degree (3)
and coeff 1,0,-3, 0. Now you may use P1 !
 
After the display of P1, you return to the polynomial main menu. Choose
DERIVATIVE option, then polynomial P1, you will soon get the result.
You may then store it in  P1, P2 or P3 using the STO menu.
Let us choose P2.
 
Back to the polynomial main menu you may now choose EQUATIONS, SIGN P2
to get sign of derivative function.
 
EX. 2 : Now you want to compute p(x)=(x-1)(x-2)-3x^2+3
and then you want to study the variation of this function.
 
Before running AASTART define this function in Y1 using [Y=] key.
Y1=(X-1)(X-2)-3X^2+3
Then run AASTART, choose P(X), DEF POLY, Y1 -> P1 and program start
P(x) expanding. You will get P(X)=-2X^2-3X+5 and you will return to
polynomial main menu. Then you may ask for function variation study using
STUDY option. (You will get the derivative function, the sign of derivative,
and the array of variation).
 
3) RATIONAL FUNCTIONS : run AASTART and choose P(X)/Q(X)
 
First you must define P and Q using DEF P, Q option. (You will have to
enter degree and coeff).
 
Then choose one of the other options of Rational functions main menu :
SIGN     : sign study of P/Q.
DERIVE   : compute the numerator P'Q-PQ' of derivative function. (denominator
           is Q^2, so no computation is needed).
SIMPLIFY : for instance (x^2-1)/(x^2+x-2) simplifies to (x+1)/(x+2).
VALUE    : rational value in one point.
STUDY    : definition set, simplification, derivative, sign of derivative,
           array of variation, asymptote.
 
4) Conditions of use
 
These programs are for educational use. They give a step by step
solution of problems, using same methods as teachers and students :
 
- Solving polynomial equation of degree 1 and 2
 
- Solving polynomial equation of degree 4 without terms in X and X^3 :
  ax^4+bx^2+c=0
 
- Solving polynomial equations of degree 3, 4, 5, 6 when it is possible
to get a solution A and then to factor (X-A) in P(X) in order to get
a lower degree equation. In these cases, as students do, program will search
for an obvious solution (-2,-1,0,1,2) and then, if not found, try to
get one solution between -10 and 10 using numerical methods. It is the only
part of these programs where numerical methods are used.
 
You may use polynomial functions of degree less than or equal to 6.
 
For P(x)/Q(x) you must make sure that the degree of intermediate results
when computing derivative function will not exceed 6, so deg(P)+deg(Q) must
be less than or equal to 7.
 
In fact, we may say these programs are able to solve most of the problems
that a student is suppose to solve with symbolic computation using
real numbers.
 
The only exception is the case of very tricky factoring process for high
degree polynomial functions.

