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Q&A #104

Teachers' Lounge Discussion: Factoring trinomials

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From: M.J.Bell <mjbell@tenet.edu>
To: Teacher2Teacher Public Discussion
Date: 1998062612:44:02
Subject: Factor Table

I was delighted to see your "tic tac toe" method of factoring
trinomials.  It is very similar to a method I devised several years
ago that I called a factor table.  I have taught this at some
conferences and found it extremely useful in the classroom.  I think
that it might be a little easier to follow than the tic tac toe, but
it is also difficult to explain in this format.  I start with a 3 x 3
table as before with one extra box on top.  I'll try to explain the
table here.  The 3 by 3 part of the table always involves
multiplication.  Multiply across and multiply down.  The d cell is the
result of adding the c and g cells.  Thus the basic rule for the
students to remember is mult. across, mult. down and add up.

          d
    a  b  c
    e  f  g 
    h  i  j
   
1. Put the first term of the trinomial in cell h.
2. Put the last term in cell i.
3. Put the middle term in cell d.
4. Multiply h and i to find j.  I call this the checking box.  
5. Look for the c and g cells.  I call these the "key cells".  
   Their product is in j and their sum is in d.  They are 
   interchangeable.
6. Although you now have a choice, I suggest that the next cell is a.
   It should be the GCF of c and h.
7. The other cells are now easy to figure out.
8. The factors are in the diagonals of 
    a  b
    e  f

Example:  Factor  6x^2 + 5x - 4
                   5x
   2x      4    8x (key)
   3x     -1   -3x (key)
   6x^2   -4   -24x^2
      
The factors are (2x - 1) and (3x + 4)
The table can be checked at a glance by multiplying across,
multiplying down, and adding up the two key cells.

This also works well for factoring 4 terms that can only be down be
grouping.  Use the 2 middle terms for the key numbers and their is no
need for an addition box.

Example:    Factor:  2x^3 - 3x^2 + 4x - 6
    x^2      -3     -3x^2
    2x        2      4x
    2x^3     -6     -12x^3

The factors are (x^2 + 2) (2x - 3)


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