AC Method Factoring

"AC method factoring" is another method of factoring you can use when working with trinomials.  That is, adding something of the form ax² + bx + c.  This can be added to my list of methods of factoring that I posted about earlier, but is specifically used for trinomial factoring.  This is a simple method that will help you to solve problems faster, and it really isn't all that complicated.  However, as with any of the factoring methods, you have to learn and understand the steps to perform it.

This factoring method requires only a few steps, so I will go over them with you now, and then show you an example or two.  To start factoring equations, we need to have a polynomial (trinomial) in the form of ax² + bx + c.  We'll start with one demo just using letters, where x is the variable and the other letters are constants:

Ax² + Bx + C

Now, here is what you do for the AC method:

1.  The first step is to multiply the A and the C constants.  Just take the two numbers (see the example that follows), and multiply them.  Keep this value in mind.

2.  Next, we want to find two values whose sum equals B, and which multiply to the product we found for AC.  This might take some trial and error, but when you find these two numbers, let's call the smaller one M and the bigger one N.

3.  Rewrite the original equation (Ax² + Bx + C) by substituting in our values for M and N.  That is, we substitute in Mx + Nx = Bx.  So then, we now have:

Ax² + Mx + Nx + C

4.  Once we have the expression in this form, we can now try to factor by grouping.  (Refer to my earlier post about methods of factoring to review grouping, if you need.  I will also be posting more about this trick soon.)

5.  Rearrange the grouped expression by the distributive property to arrive (hopefully) at your final solution!

This method of factoring will come in quite handy when you come across trinomials, especially ones that have a value for A and don't appear to have an immediately obvious way to factor or simplify.  This AC method of factoring will come in very handy in these situations!

Let's try factoring an example now with real numbers.  Follow along with the steps I've listed above.

Factor:  2x² + 9x + 9
Step 1:  AC = (2)(9) = 18
Step 2:  Two numbers to sum to 9 and multiply to 18.  Pairs can be 1 and 8, 2 and 7, 3 and 6, etc... this should be evident that the pair we like is 3 and 6.  They sum to 9 and multiply to 18.  So, M=3 and N=6
Step 3:  = 2x² + 3x + 6x + 9
Step 4:  = (2x² + 3x) + (6x + 9).  This can be factored to x(2x + 3) + 3(2x + 3).
Step 5:  Rearrange: (2x + 3)(x + 3).

And this would be your solution.  You just used the AC method to factor a trinomial!  I hope that demonstrates this method for you and that you can understand the steps you need to perform.  Let me know if you'd like more examples, and I can put some up!  As always, please +1 me below if this helped you!