
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: