Factoring and Expanding Polynomials

This shows how to check answers that require students to factor or expand a polynomial expression.

Complete Code

POD for Macro Files

PG problem file

Explanation

DOCUMENT();
loadMacros(
    'PGstandard.pl',               'PGML.pl',
    'contextLimitedPolynomial.pl', 'contextPolynomialFactors.pl',
    'contextLimitedPowers.pl',     'PGcourse.pl'
);

Preamble

In the initialization section, we need to include the macros file contextLimitedPolynomial.pl, contextPolynomialFactors.pl and contextLimitedPowers.pl.

#  Vertex form
Context("Numeric");
$n          = list_random(4, 6);
$a          = random(2, 4, 1);
$b          = ($a + $n);
$h          = ($b - $a) / 2;
$k          = $h**2 + $a * $b;
$vertexform = Compute("(x-$h)^2-$k");

#  Expanded form
Context("LimitedPolynomial-Strict");
$p0           = $h**2 - $k;
$p1           = 2 * $h;
$expandedform = Formula("x^2 - $p1 x + $p0")->reduce;

#  Factored form
Context("PolynomialFactors-Strict");
Context()->flags->set(singleFactors => 0);
LimitedPowers::OnlyIntegers(
    minPower => 0,
    maxPower => 1,
    message  => "either 0 or 1",
);
$factoredform = Compute("(x+$a)(x-$b)");

Setup

To construct this quadratic, we choose a nice factored form (x+$a)(x-$b) and from it we construct its vertex form (a(x-h)^2+k) and expanded form (ax^2+bx+c).

For the expanded form we use the LimitedPolynomial-Strict context, construct the coefficients $p[0] and $p[1] 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.

For the factored form we need to change to the PolynomialFactors-Strict context and restrict the allowed powers to either 0 or 1 using the LimitedPowers::OnlyIntegers block of code. Note: restricting all exponents to 0 or 1 means that repeated factors will have to be entered in the form k(ax+b)(ax+b) instead of k(ax+b)^2. 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 reducible quadratic factors. There are no restrictions on the coefficients, i.e., the quadratic could have any nonzero leading c oefficient. We set singleFactors=>0 so that repeated, non-simplified factors do not generate errors.

BEGIN_PGML

The quadratic expression [` [$vertexform] `] is written in vertex form.

a. Write the expression in expanded form [` ax^2 + bx + c `].

    [_____]{$expandedform}

b. Write the expression in factored form [` k(ax+b)(cx+d) `].

    [_____]{$factoredform}
END_PGML

Statement

This is the problem statement in PGML.

BEGIN_PGML_SOLUTION
Solution explanation goes here.
END_PGML_SOLUTION

ENDDOCUMENT();

Solution

A solution should be provided here.