A student of mine is preparing for the semester final in college algebra. About a third of the test is focused on factoring polynomials. To do well on the test students must have mastered the main idea of factoring trinomials into a pair of binomials. They must also be able to factor trinomials with coefficients greater than one, recognize and factor expressions which are the difference of two squares (often cleverly disguised to look more complicated than they really are), factor third degree polynomials by grouping and possibly use the formula for factoring the sum or the difference of two cubes. Anyone else think this topic is getting a bit more attention than it needs?
Don't get me wrong here, I happen to love factoring. Give me a bunch of tricky factoring problems and I'll be happy for hours. They are fun puzzles that let me play with numbers.
Being able to easily factor polynomials is occasionally useful. You can use it to solve quadratic equations (if the quadratic happens to factor nicely). Tricks like the difference of two squares show up here and there in derivations and you might have opportunities to use your factoring skills when working on problems in calculus or differential equations. I'd love to hear any other neat examples of ways factoring is useful.
The book I was looking at today has a section in its factoring exercises called 'Exercises in Application" or something close to that. It was four of the same type of exercise, each depicting squares inside of squares with some portion shaded and the lengths labeled with numbers or variables. The task, if you haven't already guessed, it is to write an expression for the shaded area and the fully factor it. Oh good! Now if I ever need to write in expression in the course of buying carpet for perfectly square surfaces I will be able to factor my expression! What a useful application...
I'd be curious to learn just how factoring polynomials got to be the focus of so much attention in our standard math pedagogy. I imagine (and this is pure speculation) that the basic task of factoring a simple trinomial happened to be difficult for some students. Teachers began inventing different ways to present factoring and to incorporate different types of factoring into their curriculum to get the point across. At some point the means sort snatched the spot-light away from the ends (perhaps helped along by the fact that many math teachers agree with me that factoring is fun!) and now we have the current situation where we insist that every college algebra student know how to factor multivariable polynomials with large coefficients.
Students who find factoring easy and fun should certainly be allowed to indulge in these nifty little puzzles. Students who find it easy but not fun shouldn't have to dwell on it once they get the main idea. But it's the students who find it difficult and tiresome that are hurt the worst. If a student has trouble with factoring, inundating them with all the different complex kinds of factoring we can come up with is not going to help them become better at factoring. Much more fundamental skills are weak or lacking. The time we spend teaching these students how to factor the sum of two cubes could be spent solidifying their basic understanding of multiplication, division, exponents and how they apply to expressions with variables.