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

The macros contextLimitedPolynomial.pl, contextPolynomialFactors.pl, and contextLimitedPowers.pl are needed for this example.

Context('Numeric');

$a = random(2, 5);
$b = ($a + 2 * random(2, 5));

#  Vertex form
$h          = ($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

Randomly generate $a and $b. The quadratic in this problem will be (x + $a)(x - $b).

Then compute the vertex form (a(x - h)^2 + k), expanded form (ax^2 + bx + c), and factored form (x + $a)(x - $b) of the quadratic in different contexts.

The vertex form is computed in the default Numeric context. This form is used for display only and will not be used as an answer. So particular care with specialized contexts is not needed.

The expanded form is computed in the LimitedPolynomial-Strict context. The coefficients $p[0] and $p[1] are constructed as Perl reals, and then the $expandedform computed using these coefficients. This is because the LimitedPolynomial-Strict context does not allow computations in the coefficients.

The factored form is computed in the PolynomialFactors-Strict context. The context is further restricted to allow only the exponents of 0 or 1 using LimitedPowers::OnlyIntegers. Note that 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. This also means that the polynomial must factor as a product of linear factors (no irreducible quadratic factors can appear). Of course, the exponents of 0, 1, or 2 could be allowed, 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 coefficient. The context flags singleFactors => 0 is set 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}{15}

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

    [_]{$factoredform}{15}
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.