The AC Method, or Factoring Trinomials using Grouping
Previously, we learned how to factor polynomials using grouping and then how to
factor quadratic trinomials with a leading coefficient of “1” by using factoring by grouping.
One important thing to remember is that when using factoring by grouping, the product of the
outer two terms equals the product of the inner two terms. This is the crux of how the “ac
method” works. First, though, a quick review by example.
Example 1: Factor x2 + 3x – 18 by using grouping.
x2 + 3x – 18
x2 _____ _____– 18
_________
____(
_________
)
(
____(
)(
)
)
Factors
of -18
-1 18
1 -18
-2
9
2
-9
-3
6
3
-6
Product of the Outer
Terms is –18x2 ,
So the product of the
inner two terms must
also be –18x2.
These add to be +3!
Notice how this answer is the same as the “shortcut” method you were taught by your past
teachers: (x – 3)(x + 6).
This method works exactly the same way for all quadratic trinomials (trinomials of the form
ax2 + bx + c, which is the reason it’s called the “ac method”. The first term times the last term
– ignoring the variables – is a•c.) The shortcut that we used above does not work if the “a”
value in the quadratic isn’t a “1”, which is what makes the “ac method” nice: it’s consistent.
Example 2: Factor 6x2 – 7x – 3.
6x2 – 7x – 3
6x2 ____x ____x – 3
_________
_________
____(
)
(
)(
____(
)
Factors
of -18
-1 18
1 -18
-2
9
2
-9
-3
6
3
-6
Product of the Outer
Terms is –18x2 ,
So the product of the
inner two terms must
also be –18x2.
)
This method also works with the difference of squares, “backwards” trinomials, and multivariable set-ups.
Difference of Squares Example: Factor 9x2 – 4.
Factors
of -36
9x2 – 4
9x2 + 0x – 4
9x2 _____ _____ – 4
_________
_________
____(
)
(
)(
____(
Product of the Outer
Terms is –36x2 ,
So the product of the
inner two terms must
also be –36x2.
)
)
“Backwards” Trinomial Example: Factor 12 + 5x – 2x2
Factors
of _____
12 + 5x – 2x2
12 _____ _____ – 2x2
_________
_________
____(
)
(
)(
____(
Product of the Outer
Terms is ________ ,
So the product of the
inner two terms must
also be ________.
)
)
Multi-Variable Trinomial Example: Factor 15x2 + 11xy + 2y2.
Factors
of _____
15x2 + 11xy + 2y2
15x2 ______ ______ + 2y2
_________
_________
____(
)
(
)(
Factor the following:
1) 6x2 – 11x – 10
____(
)
)
2) 5x2 – 7xy + 2y2
Product of the Outer
Terms is ________
So the product of the
inner two terms must
also be ________