It so happens that . And this, all by itself, far from being a cause for despair, is kind of a cool thing to know. What depresses me more than I can safely say is that “factoring trinomials” is a very big deal around here (I blog out of the office even during breaks because I don’t have the social skills needed to establish an internet connection at home) and that this particular equation very clearly has no place in our curriculum.
So. What then are we to do? Well, first of all, obviously, verify it (take nothing on authority in mathematics). Maybe you’ll notice that the RHS (“right hand side”, natch) can be parsed as ; for me this was the easiest way to check (because the Special Product Formulas of our 103 class are second nature to me; I can do the expansion to [a “difference of squares”] mentally and mentally use the “square of a binomial” formula to expand further; one “cancel”s; done). Or not; it’s fun to watch six terms of the longer expansion wipe themselves out. Then what? Well, then ask, “how could I have found this out if I didn’t know it already?”.
And the awful truth there is: Google it. This online rootfinder was pretty easy to find and gave back our result more-or-less correctly (I’m quibbling with some notations and suchlike conventions here; nonexperts needn’t be troubled by my weaselwordage). “Google it” is the right answer to a lot of questions!
No, really. Stick with me here. Lord Satan appears to you and says “Factor x^4+x^2+1 over the integers and I will spare your tribe; fail and I will annihilate you all without appeal.”. What do you do? Well, if you’re good in algebra, you solve it (and check it very carefully); if not, well, if you’ve got any sense, you get up on the internets and find a page that’ll do it for you. The last thing you’d do (if you’ve got any sense) is take our 103 course or anything like it: not only won’t they tell you how to find this out, they’ll probably leave you with the impression that you know for sure that x^4+x^2+1 is prime or something; alas, your very teacher will probably think it’s prime.
So what are we spending so much time and money for, trying to teach these poor devils to do badly what an easily-accessed robot slave can do well? And, assuming for the moment that there are good reasons for doing so, is it at least possible to be honest about it?
Because, check it out: if we rewrite the question as “How could I have found this out without a computer?”, we’ll encounter some interesting math, along with some insights into the Remediation Congame that has replaced Mathematics Education rightly-so-called with its putrefying corpse.
First of all, notice that Lord Satan said we had to factor our polynomial over the integers. But if he’d said “over C” (i.e., over ; this typesetting interface is free but otherwise far from perfect), one would (do I get to say “of course” here?) have a different answer: , where . My point here, for now, is simply that the domain over which we are to factor a given polynomial matters: it’s part of the problem and needs to be part of the problem statement (again, assuming we’re interested in trying to be honest). I’m not proposing here that 103 students need to learn about the complex number field before learning about factoring polynomials (and neither do I say that that it’s necessarily a bad idea for them to do so); one could point out that x^2 – 2 is prime over the rationals but over the reals (and, indeed, seeing that our texts and courses seem determined not to make this point, one is led to wonder how—if at all—they expect to be taken seriously in making such a fuss about the distinction between the reals and the rationals in an earlier part of the course … every chance to actually use the distinction having apparently been rigorously suppressed).
OK. Once we do allow ourselves to mention C, we can make a great deal of progress. It’s probably not wildly optimistic to hope that the median-level students of a 104 class could use the quadratic formula to write down the four complex solutions to , namely . It probably is wildly optimistic to hope that they could then write down the four roots separately. I’m not kidding. A few “A” students might be able to produce the equation
(which is of course equivalent to the C-factorization I gave earlier). A student that could produce this equation with no other instruction from me than “Factor x^4+x^2+1 over C” would gladden my heart more than I can say even though I don’t approve of square-roots-of-negatives written without the “plus-minus” sign. Bitter experience convinces me that most 104 students, even given the “x=” solution I provided above, couldn’t produce the factored-form equation with me right beside ’em telling ’em how to do it. It’s like they feel themselves physically incapable of writing down anything so messily algebraic. Not that I blame ’em (much); you’ve gotta walk before you run.
OK. But a 148 student could probably do it. Still couldn’t multiply out the result and check it, though: for that we’ll need a better notation. So let and take it from there: , , and so on (inscribing a hexagon in the unit circle is helpful here). But we sure as heck don’t do this in 148 around here; it’d take a week. And nobody cares. Not the students, not the publishers, not the department, nobody. Just me.
What am I even supposed to be doing here? Does everybody else involved actually want me to get up in front of the room and say, “Look, in real life you’re going to let the computer take over whenever anything gets at all complicated, so never mind trying to understand the big picture: that’s for the experts anyway, and the creators of this course don’t want you to learn to think like an expert …”? Because I don’t need this god-damn job that bad. I can beg on the street.