Factoring Trinomials (Quadratics?) Using The Box Method

[Ellie26]

Hey there!

I would really appreciate some help with this. I will do my best to explain it well.

I have been trying to learn to factor Trinomials (or Quadratics (I see varying terminology)) where a is greater than 1. I have found the Box Method explained here.

After I saw how those problems were worked out, I found a worksheet and attempted the method. This is one of those problems on the worksheet for an example:

8s^2 - 18s - 56

I multiplied a *times* c to get -448 and then I considered all the factors of -448:

-1, -2, -4, -7, -8, -14, -16, -28, -32, -56, -64, -112, -224, -448​

I saw -32 and 14 as the correct factors since -32 + 14 = -18 which = c
I made a box graph on my notebook paper, I will try to make a similar one here.

4s 7
^ ^
| |
2s <--[8s^2][14s]
-8 <--[-32s][-56]

And I end up with 2(x - 8)(-4 + 7)

Obviously after I FOIL it I know it is wrong since it definitely doesn't add up to the original problem. I realize I'm not doing it right, and I would appreciate guidance on how to do it right. How do I know what numbers to come out of Box, and how to arrange them in my factorized equation?

Is there an easier method that doesn't involve guessing and trial and error?

 

[mmm4444bot]

I have been trying to learn to factor Trinomials (or Quadratics (I see varying terminology)) …

First, some notes about terminology.

Trinomials are polynomials containing three terms. A trinomial might or might not be a quadratic polynomial.

Quadratic polynomials are polynomials of degree two (i.e., the largest exponent is 2). A quadratic polynomial might or might not be a trinomial.

where a is greater than 1. I have found the Box Method explained here.

I'm not familiar with the Box Method. There's another method; it's called Factoring by Grouping (shown below).

8s^2 - 18s - 56

I multiplied a *times* c to get -448 and then I considered all the factors of -448:
-1, -2, -4, -7, -8, -14, -16, -28, -32, -56, -64, -112, -224, -448

I saw -32 and 14 as the correct factors since -32 + 14 = -18 which = c

I'm sure you meant b.

… I [ended] up with 2(x - 8)(-4 + 7)

That result has some other issues.

The variable in this exercise is s, not x.

Is (-4 + 7) also a typo? That is, do you have an s in there, somewhere?

Is there an easier method that doesn't involve guessing and trial and error?

In the Factor by Grouping method, the first two steps are the same:

Multiply a*c

Find factors of a*c that sum to b

Here's how the rest goes:

Use the factors found to rewrite the given polynomial, replacing the middle term with a difference of terms.

8s^2 - 32s + 14s - 56

Next, group the first two terms and group the last two terms.

(8s^2 - 32s) + (14s - 56)

Factor the first group.

8s(s - 4) + (14s - 56)

Factor the second group, keeping in mind that we expect the same factor (s-4).

8s(s - 4) + 14(s - 4)

Now, recognize that (s - 4) can be factored out of this result because it appears on both sides of the plus sign.

(s - 4)(8s + 14)

To finish, factor 2 out of the second factor above.

2(s - 4)(4s + 7)


There are written and video lessons on-line for the Factor by Grouping method. Google keywords quadratic factor by grouping.

One last comment: I would have begun by factoring out 2 right away.

8s^2 - 18s - 56

2(4s^2 - 9s - 28)

a*c is now -112; proceed as above.

PS: When you recognize that a constant can be factored out first (like the 2), don't forget to write it, in your final answer. Many times, I have seen students report (s-4)(4s+7) because they wrote a lot of subsequent steps on their scratch paper without bothering to copy that outside factor of 2 each time. :cool:

 

[tkhunny]

x + y - 1 is a trinomial. It is not a quadratic.

The best method is the one that makes sense to you.

8s^2 - 18s - 56

It's a good idea to first factor out the common factor.

2(4s^2 - 9s - 28)

What will it look like when we are done?

2(As+B)(Cs+D)

We need A*C = 4, so find factors of 4.

1*4 or 2*2

We need B*D = -28, so find factors of 28.

1*28 or 2*14 or 4*7

That's it. Only 6 possible combinations if it is going to work.

Just try them. Think about the sign of -28. One of those factors will be negative.

With 1*4 vs 1*28
One way, we get 1 and 112. Can't get a 9 out of that.
Or 4 and 28, Can't get a 9 out of that.

With 1*4 vs 2*14
One way, we get 2 and 56. Can't get a 9 out of that.
Or 8 and 14, Can't get a 9 out of that.

With 1*4 vs 4*7
One way, we get 4 and 28. Can't get a 9 out of that.
Or 7 and 16... Hmmmm... That could lead to something.

Just be systematic.