Thread: Factoring trinomials

This method of factoring trinomials was shown to me
. . by one of my students many years ago.
You may find it as fascinating as I did.

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Example:

Write the two pairs of parentheses and "split the x's": .

Use the first coefficient twice: .


Multiply the first coefficient by the last coefficient: .

We will factor into two parts.
Note the sign of the last term.
. . If "+", think sum.
. . If "-", think difference.

Since the last sign is "+", factor into two parts
. . whose sum is the middle coefficient .

To factor into all possible pairs, divide by 1, 2, 3, ...
. . keeping those that "come out even".



The pair with a sum of 35 is: and

Since the middle coefficient is , we will use: and


Insert them into the parentheses: .

Factor out all common factors and discard them: .

Answer: .

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Example: .

We have: .


Then: .


The last term is negative, think difference.
Factor into two factors whose difference is



The pair with a difference of 11 is and

Since we want , we will use: and


Insert them into the parentheses: .

Factor out common factors and discard them: .

Answer: .

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At this point, someone will say, "Why not use the Quadratic Formula?"
. . Why not, indeed?
In fact, why bother teaching Factoring at all?
. . And I don't have an answer to that question.

But it would be a shame to run through the Quadratic Formula
. . to factor, say, or

It reminds me of the "box method"

If you have a trinomial, then write out a box like this:

Code:

:-----:-----:
:  a  :     :
:-----:-----:
:     :  c  :
:-----:-----:

Now you need to fill in the two empty squares. The trick is that they add to "b" and multiply to "ac", I'll use as an example:

Code:

:-----:-----:
:  2  :  2  :
:-----:-----:
:  3  :  3  :
:-----:-----:

Then find the greatest common factor of each row and collum:

Code:

     1      1
  :-----:-----:
2 :  2  :  2  :
  :-----:-----:
3 :  3  :  3  :
  :-----:-----:

So: