View of /trunk/NationalProblemLibrary/Rochester/setAlgebra23PolynomialZeros/descartes2.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('Decartes rule of signs','real roots')
    3 ##TYPE('word problem')
    4 ##ENDDESCRIPTION
    5 
    6 ## Tagged by sawblade
    7 
    8 ## DBsubject('Algebra')
    9 ## DBchapter('Polynomial and Rational Functions')
   10 ## DBsection('Real Zeros of Polynomials')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGauxiliaryFunctions.pl",
   28 "PGpolynomialmacros.pl"
   29 );
   30 
   31 TEXT(beginproblem());
   32 $showPartialCorrectAnswers = 0;
   33 
   34 
   35 $degree = random(6,9,1);
   36 
   37 foreach $i (0..$degree) {
   38         $poly[$i] = non_zero_random(-20,20,1);
   39 }
   40 
   41 ($maxpos,$maxneg) = Descartes(~~@poly);
   42 
   43 $poly = PolyString(~~@poly);
   44 
   45 BEGIN_TEXT;
   46 Use Descartes' Rule of Signs to analyze the number of positive and negative real
   47 roots and the number of non-real roots of the function:
   48 \[h(x) = $poly\]
   49 $BR
   50 $BR
   51 There are at least \{ans_rule(5)\} and at most \{ans_rule(5)\} positive real
   52 roots.
   53 $BR
   54 There are at least \{ans_rule(5)\} and at most \{ans_rule(5)\} negative real
   55 roots.
   56 $BR
   57 There are at least \{ans_rule(5)\} and at most \{ans_rule(5)\} non-real
   58 roots.
   59 $BR
   60 END_TEXT
   61 
   62 if (int($maxpos/2)==$maxpos/2) {$minpos = 0;}
   63 else {$minpos = 1;}
   64 if (int($maxneg/2)==$maxneg/2) {$minneg = 0;}
   65 else {$minneg = 1;}
   66 $mincomp = $degree-($maxpos+$maxneg);
   67 $maxcomp = $degree-($minpos+$minneg);
   68 
   69 ANS(num_cmp($minpos));
   70 ANS(num_cmp($maxpos));
   71 ANS(num_cmp($minneg));
   72 ANS(num_cmp($maxneg));
   73 ANS(num_cmp($mincomp));
   74 ANS(num_cmp($maxcomp));
   75 
   76 
   77 ENDDOCUMENT();        # This should be the last executable line in the problem.
   78