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This is factoring page 3c.   It helps you FACTOR TRINOMIALS

In the first page, I presented the perfect square trinomial and the difference of two squares.

In the second page, I presented a method of factoring four-term expressions with pairs of factors.

Pages 3a, 3b, 3c, and 3d give three methods of factoring trinomials:

Page 3a works by listing the factors of the "a" and "c" coefficients.
Page 3b completes the square.
Page 3c (this page) is the "AC Method", which has you multiply the "a" and "c" coefficients, and list all the factors of that product.
Page 3d is the "Simplified AC Method"

Some people say this is the easiest method of all.  I present it last so you'll have an appreciation for it, once you've seen the other methods!  

This method is sometimes referred to as the "AC method", because you multiply the "a" and "c" coefficients, and then find all the factors of the product.

How to Factor Using the AC Method

First, consider the general form of the quadratic expression:

ax� + bx + c

Now look again at EXAMPLE 5:

6x² + 19x - 20

The "a" coefficient is 6, and the "c" coefficient is 20.

Multiply the 6 and the -20 together to get -120.

Next, WRITE DOWN all of the ways you can multiply 2 integers together to get -120.

Remember, if a number has 16 factors, there are 16 different ways to multiply two integers together to get that number (because 16 is even).  Don't forget the negative numbers!  For example, if 3 times 4 is 12, then -3 times -4 also equals 12.

Now here are the 16 ways to multiply two integers together to get -120.

 -1, 120   -120, 1
 -2, 60   -60, 2 
 -3, 40   -40, 3 
 -4, 30   -30, 4 
 -5, 24   -24, 5 
 -6, 20   -20, 6 
 -8, 15   -15, 8 
 -10, 12   -12, 10 

From this list, look for the combination that ADDS up to the "x" term -- the "b" coefficient -- in the original expression, in this case, 19. From the list, you can see that the fifth line has -5 and 24 which add up to 19.

Rewrite the original expression, splitting up the 19x term into 2 terms:

-5x and 24x

(Note: the order doesn't matter -- see for yourself this will work if you write these two terms as 24x and -5x instead.)

6x� - 5x + 24x - 20

Next, factor by grouping the first two terms and the second two terms:

(6x� - 5x) + (24x - 20)
x(6x - 5) + 4(6x - 5)

Now, factor out the terms in parentheses:

(x+4) (6x - 5)

You get the same answer regardless of which way you split up the "x" term.  If you doubt that, try factoring

6x� + 24x - 5x - 20

and you'll see you get the same answer.

This method was shared on the message board by Steve Bast.  Steve says,

"Ever since I saw this "AC method" in a college algebra textbook by Bello (1977), I never use other "guessing" methods any more. For those of you who are teachers, I think this is a much better way to teach factoring rather than guessing.

If you weren't able to factor a quadratic equation using any of the methods in this section, then either it can't be factored, or else the factors are not rational numbers.  In either case, you'll find out what's going on using the QUADRATIC FORMULA -- that's the topic of the next factoring page.

Related pages in this website

Next: the Simplified AC Method

The Locker Puzzle -- if student n reverses the door position of every n'th locker, how many will be left open?

Highly Composite Numbers -- a discussion of the magical features of numbers that have a whole slew of factors.


The webmaster and author of this Math Help site is Graeme McRae.