Factoring most famously comes up in (a) limits of fractions with canceling terms and (b) partial fractions. In doing problems in the category of (a) early in the year, students in my class found factoring polynomials something to bond over. In particular, they talked about how horrible synthetic division is and how annoying long division is. I usually give a little spiel about the method of unmultiplying, which many students ignore but a handful find revolutionizes their relationship with factoring.
(Unmultiplying is essentially short division in your head. Say you want to divide x^3 + 8 by x+2. So we write (x+2). First you know you need an x^2, so we have (x+2)(x^2 …. Now that gives us an extra 2x^2 term which we need to cancel, so we add a -2x term: (x+2)(x^2 - 2x …. That leaves an extra -4x term, so we add a 4 to cancel it: (x+2)(x^2 - 2x + 4), which gives the 8 we wanted. This method is more evocative/comprehensible explained out loud at a blackboard.)
The things I tell my students are: if you plug a number a into a polynomial and you get zero, (x-a) will divide evenly into it. In the case of limits, the root a is usually But then I have to remind them that this only makes sense for polynomials (i.e. natural number exponents). If they see fractional exponents, to try to substitute some variable for one of the fractional exponents to turn it into a polynomial situation.
In the case of partial fractions, it’s sometimes not as easy to see. I tell students the only method is to guess. If your curriculum allows it, a computing device might help to guess. In a test-taking situation, if you see a wicked polynomial you want to factor, try plugging in 1, 2, -1, -2 and see if any of them are roots. In a pinch try 3 and -3. This last is purely a test survival skill, and an unspoken convention in tests without calculators. People who are good at tests are so used to it that it seems natural, when in fact it’s very common to have students get stumped and say, “how were we supposed to know to just plug numbers in?” I think it’s only fair to have a hint along the lines of “try plugging in a few values” sometime in the semester.
Some people think that factoring is one of the skills that will disappear like slide-rule proficiency, log table reading and square root extraction in the face of technological advances. I suspect they’re right, though I leave the discussion to elsewhere whether its passing should be mourned or celebrated.
– Eric Hsu 1998-08-09