There are of course at least a dozen methods available to factor quadratic trinomials (some specific -- like difference of perfect squares -- some general, some devious -- like use the quadratic formula to find the roots and work backwards -- some paper and pencil some electronic). That implies that no one of them is that important.
What is important, IMHO or maybe just IMO, is that the student understand that the factored and expanded form of a polynomial are both important and useful; that each tell us different important and useful things about the polynomial. But the procedure of going from the one to the other (especially the expanded form to the factored form) is not important. It's a job for a CAS. Okay, sure, have your students do a few simple ones to get a "feel" for (not the procedure, but) the relationship between the two forms. Days of instruction of "factoring" are largely a waste of time. The time would be better spent in using the factored from to learn about the polynomial (graph, roots, general shape, increasing, decreasing, etc.). I know that's a big step for some. If you accept that a CAS can produce the factors, then you can study, not just quadratic polynomials, but a much larger set of polynomials and polynomials of higher degree. You can deal with polynomials that factor over larger sets than just the integers (irrational, then complex numbers).
Another thing to consider is just how few quadratic polynomials actually factor. For example if you limit the coefficients to relatively prime integers between -20 and 20, only about 7% factor over the integers. The percentage that factor is inversely proportional to the size of the coefficients. As the coefficients increase in absolute value the percentage that factor approaches zero.
Study polynomials, let the CAS do the factoring and expanding.
Lin McMullin Niantic, CT.
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