Difference between revisions of "ExpandedPolynomial1"

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(Created page with '<h2>Polynomial Multiplication (Expanding)</h2> <p style="background-color:#eeeeee;border:black solid 1px;padding:3px;"> This PG code shows how to require students to expand poly…')
 
(add historical tag and give links to newer problems.)
 
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{{historical}}
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<p style="font-size: 120%;font-weight:bold">This problem has been replaced with [https://openwebwork.github.io/pg-docs/sample-problems/Algebra/ExpandedPolynomial.html a newer version of this problem]</p>
  +
  +
 
<h2>Polynomial Multiplication (Expanding)</h2>
  
<h2>Polynomial Multiplication (Expanding)</h2>
   
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<p style="background-color:#eeeeee;border:black solid 1px;padding:3px;">
  
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[[File:ExpandedPolynomial1.png|300px|thumb|right|Click to enlarge]]
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<p style="background-color:#f9f9f9;border:black solid 1px;padding:3px;">
 
This PG code shows how to require students to expand polynomial multiplication.
  
This PG code shows how to require students to expand polynomial multiplication.
−
<ul>
  
−
<li>Download file: [[File:ExpandedPolynomial1.txt]] (change the file extension from txt to pg when you save it)</li>
  
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<li>File location in NPL: <code>NationalProblemLibrary/FortLewis/Authoring/Templates/Algebra/ExpandedPolynomial1.pg</code></li>
  
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</ul>
  
 
</p>
  
</p>
  +
<!--* File location in OPL: [https://github.com/openwebwork/webwork-open-problem-library/blob/master/OpenProblemLibrary/FortLewis/Authoring/Templates/Algebra/ExpandedPolynomial1.pg FortLewis/Authoring/Templates/Algebra/ExpandedPolynomial1.pg] -->
  +
* PGML location in OPL: [https://github.com/openwebwork/webwork-open-problem-library/blob/master/OpenProblemLibrary/FortLewis/Authoring/Templates/Algebra/ExpandedPolynomial1_PGML.pg FortLewis/Authoring/Templates/Algebra/ExpandedPolynomial1_PGML.pg]
   
  +
<br clear="all" />
 
<p style="text-align:center;">
  
<p style="text-align:center;">
 
[[SubjectAreaTemplates|Templates by Subject Area]]
  
[[SubjectAreaTemplates|Templates by Subject Area]]
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<tr valign="top">
  
<tr valign="top">
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<th> PG problem file </th>
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<th style="width: 50%"> PG problem file </th>
 
<th> Explanation </th>
  
<th> Explanation </th>
 
</tr>
  
</tr>
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<pre>
  
<pre>
 
DOCUMENT();
  
DOCUMENT();
−
loadMacros(
  
  +
−
"PGstandard.pl",
  
  +
loadMacros('PGstandard.pl','MathObjects.pl','contextLimitedPolynomial.pl',
−
"MathObjects.pl",
  
  +
'AnswerFormatHelp.pl','PGML.pl','PGcourse.pl');
−
"contextLimitedPolynomial.pl",
  
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"contextPolynomialFactors.pl",
  
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"contextLimitedPowers.pl",
  
−
);
  
   
 
TEXT(beginproblem());
 
TEXT(beginproblem());
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<p>
  
<p>
 
<b>Initialization:</b>
  
<b>Initialization:</b>
−
We need all of these macros.
  
  +
We must load <code>contextLimitedPolynomial.pl</code>
 
</p>
  
</p>
 
</td>
  
</td>
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#
  
#
 
Context("Numeric");
  
Context("Numeric");
−
$n = list_random(4,6);
  
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$h = 3;
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$a = random(2,4,1);
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$k = 5;
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$b = ($a+$n);
  
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$h = ($b-$a)/2;
  
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$k = $h**2+$a*$b;
  
 
$vertexform = Compute("(x-$h)^2-$k");
  
$vertexform = Compute("(x-$h)^2-$k");
   
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#
  
#
 
Context("LimitedPolynomial-Strict");
  
Context("LimitedPolynomial-Strict");
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$p[0] = $h**2 - $k;
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$b = -2 * $h;
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$p[1] = 2*$h;
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$c = $h**2 - $k;
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$expandedform = Formula("x^2 - $p[1] x + $p[0]")->reduce;
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$expandedform = Formula("x^2 + $b x + $c")->reduce();
−  
−
#
  
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# Factored form
  
−
#
  
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Context("PolynomialFactors-Strict");
  
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Context()->flags->set(singleFactors=>0);
  
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LimitedPowers::OnlyIntegers(
  
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minPower => 0, maxPower => 1,
  
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message => "either 0 or 1",
  
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);
  
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$factoredform = Compute("(x+$a)(x-$b)");
  
 
</pre>
  
</pre>
 
</td>
  
</td>
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<p>
  
<p>
 
<b>Setup:</b>
 
<b>Setup:</b>
−
To construct this quadratic, we choose a nice factored form <code>(x+$a)(x-$b)</code> and from it we construct its vertex form (a(x-h)^2+k) and expanded form (ax^2+bx+c).
  
  +
The macro <code>contextLimitedPolynomial.pl</code> provides two contexts:
  +
<pre>
  +
Context("LimitedPolynomial");
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Context("LimitedPolynomial-Strict");
  +
</pre>
  +
The strict version does not allow any mathematical operations within coefficients, so <code>(5+3)x</code> must be simplified to <code>8x</code>.
  +
For more details, see [http://webwork.maa.org/pod/pg/macros/contextLimitedPolynomial.html contextLimitedPolynomial.pl]
 
</p>
  
</p>
 
<p>
  
<p>
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For the expanded form we use the <code>LimitedPolynomial-Strict</code> context, construct the coefficients <code>$p[0]</code> and <code>$p[1]</code> as Perl reals, and then construct <code>$expandedform</code> using these pre-computed coefficients. This is because the LimitedPolynomial-Strict context balks at answers that are not already simplified completely.
 +
We use the <code>LimitedPolynomial-Strict</code> context, construct the coefficients <code>$b</code> and <code>$c</code> as Perl reals, and then construct <code>$expandedform</code> using these pre-computed coefficients. This is because the LimitedPolynomial-Strict context balks at answers that are not already simplified completely. Notice that we called the <code>-&gt;reduce()</code> method on the expanded form of the polynomial, which will ensure that the polynomial will be displayed as <code>x^2 - 6x + 4</code> instead of <code>x^2 + -6x + 4</code>.
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</p>
  
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<p>
  
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For the factored form we need to change to the <code>PolynomialFactors-Strict</code> context and restrict the allowed powers to either 0 or 1 using the <code>LimitedPowers::OnlyIntegers</code> block of code. Note: restricting all exponents to 0 or 1 means that repeated factors will have to be entered in the form <code>k(ax+b)(ax+b)</code> instead of <code>k(ax+b)^2</code>. Also, restricting all exponents to 0 or 1 means that the polynomial must factor as a product of linear factors (no irreducible quadratic factors can appear). Of course, we could allow exponents to be 0, 1, or 2, but then students would be allowed to enter <i>reducible</i> quadratic factors. There are no restrictions on the coefficients, i.e., the quadratic could have any nonzero leading coefficient. We set <code>singleFactors=>0</code> so that repeated, non-simplified factors do not generate errors.
  
 
</p>
  
</p>
 
</td>
  
</td>
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<td style="background-color:#ffdddd;border:black 1px dashed;">
  
<td style="background-color:#ffdddd;border:black 1px dashed;">
 
<pre>
  
<pre>
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Context()->texStrings;
  
  +
BEGIN_PGML
−
BEGIN_TEXT
  
  +
The quadratic expression [` [$vertexform] `]
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The quadratic expression \( $vertexform \)
  
  +
is written in vertex form. Write the
−
is written in vertex form.
  
  +
expression in expanded form [` ax^2 + bx + c `].
−
$BR
  
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−
$BR
  
  +
[_____________________]{$expandedform}
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(a) Write the expression in expanded form
 
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−
\( ax^2 + bx + c \).
  
  +
[@ helpLink('formulas') @]*
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$BR
  
  +
END_PGML
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\{ ans_rule(30) \}
  
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$BR
  
−
$BR
  
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(b) Write the expression in factored form
  
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\( k(ax+b)(cx+d) \).
  
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$BR
  
−
\{ ans_rule(30)\}
  
−
END_TEXT
  
−
Context()->normalStrings;
  
 
</pre>
  
</pre>
 
<td style="background-color:#ffcccc;padding:7px;">
  
<td style="background-color:#ffcccc;padding:7px;">
 
<p>
  
<p>
 
<b>Main Text:</b>
  
<b>Main Text:</b>
−
Everything here is as usual. To help students understand how to format their answers, we give examples <code>ax^2+bx+c</code> and <code>k(ax+b)(cx+d)</code> of what the answers should look like.
 +
To help students understand how to format their answers, we give an example <code>ax^2+bx+c</code> of what the answer should look like.
 
</p>
  
</p>
 
</td>
  
</td>
 
</tr>
  
</tr>
−  
−
<!-- Answer section -->
  
−  
−
<tr valign="top">
  
−
<td style="background-color:#eeddff;border:black 1px dashed;">
  
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<pre>
  
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$showPartialCorrectAnswers = 1;
  
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ANS( $expandedform->cmp() );
  
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ANS( $factoredform->cmp() );
  
−  
−
</pre>
  
−
<td style="background-color:#eeccff;padding:7px;">
  
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<p>
  
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<b>Answer Evaluation:</b>
  
−
Everything is as expected.
  
−
</p>
  
−
</td>
  
−
</tr>
  
−  
   
 
<!-- Solution section -->
  
<!-- Solution section -->
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<td style="background-color:#ddddff;border:black 1px dashed;">
  
<td style="background-color:#ddddff;border:black 1px dashed;">
 
<pre>
  
<pre>
−  
  +
BEGIN_PGML_SOLUTION
−
Context()->texStrings;
  
−
BEGIN_SOLUTION
  
−
${PAR}SOLUTION:${PAR}
  
 
Solution explanation goes here.
  
Solution explanation goes here.
−
END_SOLUTION
  
  +
END_PGML_SOLUTION
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Context()->normalStrings;
  
−  
−
COMMENT('MathObject version.');
  
   
 
ENDDOCUMENT();
  
ENDDOCUMENT();
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[[Category:Top]]
  
[[Category:Top]]
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[[Category:Authors]]
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[[Category:Sample Problems]]
  +
[[Category:Subject Area Templates]]

Latest revision as of 04:47, 18 July 2023

This article has been retained as a historical document. It is not up-to-date and the formatting may be lacking. Use the information herein with caution.

This problem has been replaced with a newer version of this problem


Polynomial Multiplication (Expanding)

Click to enlarge

This PG code shows how to require students to expand polynomial multiplication.


Templates by Subject Area

PG problem file Explanation

Problem tagging data

Problem tagging: 

DOCUMENT();

loadMacros('PGstandard.pl','MathObjects.pl','contextLimitedPolynomial.pl',
  'AnswerFormatHelp.pl','PGML.pl','PGcourse.pl');

TEXT(beginproblem());

Initialization: We must load contextLimitedPolynomial.pl 

#
#  Vertex form
#
Context("Numeric");
$h = 3;
$k = 5;
$vertexform = Compute("(x-$h)^2-$k");

#
#  Expanded form
#
Context("LimitedPolynomial-Strict");
$b = -2 * $h;
$c = $h**2 - $k;
$expandedform = Formula("x^2 + $b x + $c")->reduce();

Setup: The macro contextLimitedPolynomial.pl provides two contexts: 

Context("LimitedPolynomial");
Context("LimitedPolynomial-Strict");

The strict version does not allow any mathematical operations within coefficients, so (5+3)x must be simplified to 8x. For more details, see contextLimitedPolynomial.pl

We use the LimitedPolynomial-Strict context, construct the coefficients $b and $c as Perl reals, and then construct $expandedform using these pre-computed coefficients. This is because the LimitedPolynomial-Strict context balks at answers that are not already simplified completely. Notice that we called the ->reduce() method on the expanded form of the polynomial, which will ensure that the polynomial will be displayed as x^2 - 6x + 4 instead of x^2 + -6x + 4. 

BEGIN_PGML
The quadratic expression [` [$vertexform] `]
is written in vertex form.  Write the
expression in expanded form [` ax^2 + bx + c `].

[_____________________]{$expandedform}

[@ helpLink('formulas') @]*
END_PGML

Main Text: To help students understand how to format their answers, we give an example ax^2+bx+c of what the answer should look like. 

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution:

Templates by Subject Area

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