| The
AC Method for Factoring Trinomials. |
Factor:  |
1. Multiply
the leading coefficient (the number in front of x2
) times the constant term (the last term). Remove the
leading coefficient.
(Obviously an illegal algebraic move!) |
 |
| 2. Factor this
new trinomial. |
 |
| 3. Replace
x with the original leading coefficient times x. In this example, the leading coefficient
is 2. |
 |
| 4. Factor. |
 |
| 5. Divide by
the original leading coefficient. |
 |
 |
How can an "illegal" move
result in a correct answer? Is this
really a valid method of factoring?
And if so, why does it work? |
Why it works:
The secret to understanding this
method is to realize that these steps are actually a shorthand for a
more complex process of multiplication and replacement.
Let's take one more look at the whole process. The
AC Method is simply a shorthand version of the
following procedure.
| Multiply the entire
expression times the leading coefficient: |
 |
| Distribute: |
|
| Re-write: |
|
Replace 2x
with another variable, such as m,
where 2x = m: |
|
| Factor: |
|
| Replace m
with 2x: |
|
| Factor: |
|
|
These factors now equal
what we started with, which is TWICE the original problem:
=
|
Divide both sides
by 2:
=  |
