Here's something I think I see a few times in any college algebra class: a really weird way of accomplishing quadratic factoring. (More generally, this might go in a larger file of "things students swear are taught by other instructors which are semi-insane" -- including whacked-out order-of-operations, keep-change-change for negatives, the idea that -4

^{2} means (-4)(4), etc.).

Anyway, let's say we want to factor what I call a "hard quadratic", i.e., Ax

^{2}+Bx+C, in integers, with A≠1 (hence "hard"). I prefer the method of grouping: i.e., factor AC so it sums to B, use those factors to split the term Bx, and then factor the four terms by grouping. Pretty straightforward.

But here's what a few students will insist on doing every semester: (1) Find factors of AC that sum to B; call these factors u & v (so that step is the same); (2) Write the expression (Ax+u)(Ax+v); (3) Look for a GCF of A in one of those binomials and strike it out.

Here's an example: Factor 5x

^{2}+7x−6.

Step (1): Note AC = −30 = (10)(−3), factors which sum to B = 7.

Step (2): Write (5x+10)(5x−3)

Step (3): Divide the first binomial by 5, producing (x+2)(5x−3).

So while this procedure does produce the right answer, what irks me tremendously is that the expression written in step (2) is not actually equal to either the original expression or the answer at the end. (Compounding this issue, students will nonetheless usually write equals signs by rote on either side of it.) Riffs on this procedure would be to write something like this on sequential lines, if you can follow it:

5x

^{2}+7x−6 → x

^{2}+7x−30 → (5x+10/5)(5x−3/5) → (x+2)(5x−3)

Again, the primary grief I have over this is that none of these expressions are equal to any of the others, and the students using this procedure are always oblivious to that fact. Second issue: They're likely to trip up over a non-elementary problem where the factor A does not appear in either of the binomials, e.g.: 4x

^{2}+4x+1 = (2x+1)(2x+1). Third issue: If there's a GCF in the quadratic itself and you overlook that, the standard grouping technique will still work (even if it's not the easiest way to do it), whereas I suspect users of this technique will be prone to incorrectly striking out any GCFs they discover at the end of the process.

Now, technically you could modify this and turn it into a correct procedure this way: Note that for quadratic Ax

^{2}+Bx+C, values u & v satisfy uv=AC and u+v=B if and only if Ax

^{2}+Bx+C = 1/A(Ax+u)(Ax+v). (Proof: 1/A(Ax+u)(Ax+v) = 1/A(A

^{2}x

^{2} + (u+v)Ax + uv) = Ax

^{2} + (u+v)x + uv/A and equate coefficients). So you could find u & v as usual, then write this latter expression, and simplify. The 1/A does always cancel out, but I've never seen a student actually write that factor in the second step.

So what I always do if I see this on a test in my college algebra class is to take half credit off for the problem and note that the intermediary expression is "false", i.e., not equal to what comes before or after. This then becomes an opportunity to discuss with the student why that's improperly written math -- went well in my most recent semester, but I can easily see that becoming more combative in a remedial algebra class.

Have you seen this (common) faulty factoring procedure in your classes? What do you as a correction for it, if anything?