It was a triple nested loop. not that much code actually, including plotting the results.

With the results plotted as percent factorable vs. largest absolute value of coefficient (i.e. x = 1, 2, 3, …20) the results approximated a corner hyperbola (k/x) approaching the x-axis as x got larger. At coefficients of 20 and less, only about 7% factored; and dropping fast.

Unfortunately, I’m not a programmer and not that familiar with mathematica either, so I’m hesitant to try to reproduce the result with Mathmatica 6. If I find the time …

Hope this helps.

Lin

1) The most powerful feature of factoring is not the process itself but the fact that a polynomial can be broken down into linear factors (over the complex numbers).

2) An inordinate amount of time is spent on teaching factoring techniques, but as you can see from the grid, very few are factorable over the rationals. And it’s worse once you go to higher degree polynomials.

Much better to spend some time factoring, then after covering quadratic functions, recap factoring in general with connection to graphs and zeros. Factor “backwards” from the zeros of the graph. GeoGebra can be a big help there. (I have a few activities of this ilk in my web site, under GeoGebra)

3) Even when the quadratic formula gives irrational or complex roots, the quadratic expression is still factorable :(x-c_1)(x-c_2). Here, c_1 and c_2 can be any real number. However, most algebra courses don’t even go there.

By the way, a small correction to one of the comments – if the parabola does not have any real zeros – it is not factorable over the real numbers. However, it is factorable over the complex numbers – just use the quadratic formula to find the roots, c_1 and c_2.
–Reva Narasimhan
Assoc. Prof. of Math , Kean University, NJ

To keep them (and myself, frankly) interested at all skill levels, I’ve been focusing on graphical factoring. I don’t have a writeup ready, but check out the ggb file embedded at http://larkolicio.us/blog/?p=50 , and take a look at the NCTM illuminations worksheets at http://illuminations.nctm.org/LessonDetail.aspx?ID=L282 . I’ve been having kids explore the properties of factors – what happens when a factor is zero, and what happens if factors of a quadratic are different signs, etc. I like this approach. I flew over some methods of factoring like differences of squares, etc, and then set kids loose on either 1) tons of factoring practice, including practice distributing for students with especially low skills, or 2) studying the significance of factors graphically and numerically for the kids who can already factor cubic polynomials analytically.

The kids exploring factors graphically are not limited to pre-made problems. And they are automatically discovering things like the fact that parabolas with no roots must not have any factors (made my day).

Please do proofread the NCTM worksheets before using them (I got a few nasty surprises during class).

i just found out last week that the new integrated algebra exam doesn’t even present kids with unfactorable quadratics. they aren’t required to know the quadratic formula, or use it on the exam.

so that’s just crazy.

i thought this post was awesome, sam. thanks so much!

This isn’t the first time people have wondered if we should teach factoring. There was a trend in the 80s to do away with it and supposedly there were understanding problems later so it got put back in.

Some countries don’t (now and historically) bother with it and just teach the quadratic formula.