Shortcuts vs Process

 

The other day, I read something on twitter asking whether one should teach only guess and check rather than shortcuts or both.  I was surprised to find out that much of what I teach to students to help them factor trinomials and some polynomials are classified as shortcuts.  I seldom actually teach guess and check because so many students find that method frustrating and they shut down.  Even I get frustrated if I have to rely on the guess and check method for factoring.

Once I learned to do the diamond method, I found it so much easier.  If you are factoring a trinomial with a leading coefficient of one, you figure out what two numbers produce a product of the third term and a sum of the second term.  The two numbers you come up with, makes up the second term of the binomials.

If you look at the equation x^2 + 2x - 15, you look at the factors that make -15 but when added together gives 2 so x^2 + 2x - 15 factors into (x -3)(x+5) On the other hand, if you have something like 2x^2 + 5x + 3, the method I use is more like the reverse of multiplying two binomials but use the diamond.

I still have students multiply the leading coefficient by the third term, the constant so you’d so 2 x3 to get 6 so what two numbers produce 6 when multiplied but add up to 5 which are 2 and 3.  The next step is to replace the 5x with 2x + 3x so you have four terms.  From there you use the grouping method to place the four terms into two groups with two terms.  Factor things out and go from there.  I have a complete step by step photo of it.  

So is this a process or is it a shortcut?  Personally I see it as a process because there are steps you follow to go from the trinomial to two binomials which are factors.  These methods also appear in many textbooks so does that validate them or are they still considered shortcuts?  Honestly, this process works if a trinomial can be factored otherwise, you end up using the quadratic formula.

I’ve also been able to use the group four terms into two groups with certain polynomials but not all.  In fact, it is one of the things I try with polynomials to the 4th degree since some of them can be factored this way.  Again, if I can apply it to  multiple situations and it’s a process that can be followed each time, it seems to me it becomes a method rather than a shortcut.  Most things, I think of as a shortcut is something like you can’t divide by a fraction so you flip the second term and multiply.  Let me know what you think, I’d love to hear.