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Factoring
Posted:
Jul 30, 1998 1:11 PM
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Here is a very useful method of factoring. Sorry, this is not
formatted very well here. If you would like a better format, send me
an e-mail request and I will send an attachment. If you find this
method useful, I would love to hear from you. mjbell@gte.net
FACTOR TABLES
By Mary Jane Bell
OBJECT:
The object is to present students with an alternative to
trial-and-error factoring techniques for quadratic trinomials. The
method is also valuable for factoring expressions with 4 terms that
can be factored into two binomials. The table becomes a fun ÃÂpuzzleÃÂ
that provides students with a clear-cut starting place and sequential
steps and an immediate check on their work.
VERTICAL TEAMING (Pre-algebra)
The object in a pre-algebra class is to practice multiplying and
factoring integers in a fun ÃÂpuzzleÃÂ and with a format that can work
easily into algebraic factoring.
TABLE LAYOUT AND RULES
Start with a 3 x 3 table with one extra box on top.
The basic operational rules are:
MULTIPLY ACROSS ÃÂ MULTIPLY DOWN ÃÂ ADD UP (to the top box)
_______________________________
| Factor Table | 18 |
---------------------------------
| 3 | 2 | 6 |
---------------------------------
| 4 | 3 | 12 |
---------------------------------
| 12 | 6 | 72 |
---------------------------------
By strategically leaving blanks, the table becomes a puzzle. You may
add signs and variables as the students are ready for them. (I
usually start with pre-algebra examples to get the students familiar
with the operation of the table.)
PRE-ALGEBRA example #1:
_______________________________
| Factor Table | - 5 |
--------------------------------
| | | | < Key
--------------------------------
| | | | < Key
--------------------------------
| - 4 | 6 | | < Checking Box
---------------------------------
1. Start by filling in the ÃÂChecking BoxÃÂ. Remember: MULTIPLY ACROSS
2. Find the two ÃÂKey BoxesÃÂ in any order. Remember that the product
of the 2 Keys is on the bottom and the sum is on the top. So you are
looking for two numbers whose product is ÃÂ 24 and whose sum is ÃÂ 5.
3. The rest of the table is easy. You may choose which of the four
remaining ÃÂFactor BoxesÃÂ to do next. Whichever is chosen should be
the GCF of the product of that row and column. (If you donÃÂt use the
GCF, you may get into fractions, which can also be an interesting and
worthwhile exercise for pre-algebra students.) You have some options
with signs. If you use some large numbers, you may want to let the
students use calculators to find the key numbers.
Solution to example #1
__________________________________
| Factor Table | - 5 |
-----------------------------------
| 4 | - 2 | - 8 | < Key
-----------------------------------
| 1 | 3 | 3 | < Key
-----------------------------------
| - 4 | 6 | - 24 | < Checking Box
------------------------------------
ALGEBRA TABLE: Example #2:
After the students understand the basic operation of the table they
are ready for the following algebraic example.
________________________________
| Factor Table | - 7x |
--------------------------------
| | | |
--------------------------------
| | | |
--------------------------------
| 6x^2 | - 3 | |
--------------------------------
Solution to example #2:
The boxes are marked (in parentheses) in the order completed, but the
students certainly have some choices. The table is quick and easy to
check at a glance in any direction.
___________________________________________
| Factor Table | - 7x |
-------------------------------------------
|(4) 3 x | (5) - 3 | (2) - 9 x |
-------------------------------------------
|(6) 2 x | (7) 1 | (3) 2 x |
-------------------------------------------
| 6x^2 | - 3 | (1) -18x^2 |
-------------------------------------------
What the students donÃÂt know is that they just factored their first
trinomial.
6x^2 ÃÂ 7x ÃÂ 3 = (3x + 1)(2x ÃÂ 3)
Note that the factors are found by taking the diagonals of the last
four boxes in the table. If the students have any doubts, they can
multiply it out and observe where each term in the table fits into the
product. This is not a problem you would usually attempt as the first
try at factoring a trinomial, but it is possible with the factor
table. Most classes of beginning algebra should be able to do this in
one long block period or maybe 2 shorter periods.
The main thing the students need to remember is where to put the
terms. If a trinomial is in standard order, ax^2 + bx + c , then the
first and last terms must go in the first 2 boxes on the bottom row
and the middle term always goes at the top. The factors are always
the diagonals of the four ÃÂFactor BoxesÃÂ in the upper left corner.
LAYOUT OF THE TRINOMIAL TABLE
__________________________________
| Factor Table | middle |
----------------------------------
| factor | factor | key |
----------------------------------
| factor | factor | key |
----------------------------------
| first | last | checking |
----------------------------------
SPECIAL CASE FACTORING
One nice thing about the Table that it works equally well on perfect
square trinomials and the difference of two squares (Let the middle
term = 0.) and the students quickly see the pattern for these special
cases that allows them to do them without the table. If the leading
coefficient is 1, most students see that the checking number is the
same as the last term and all they need are the key numbers for the
last term and the middle coefficient and many students begin to do
these in their head fairly rapidly. It also works exceptional well
with trinomials that are prime over the integers. They quickly see
that they cannot find the 2 key numbers. If the checking number is
large, they may need to use a calculator to try different key factors.
FACTORING EXPRESSIONS WITH 4 TERMS
This is one of my favorite uses of factor table. These are the
expressions that normally are factored by grouping. That has always
been a difficult procedure for my students to understand and then to
execute correctly. It is now a breeze. Note this technique only
works for expressions that will factor into two binomials.
Example #3
1. Factor: 3x^3 + 6x^2y ÃÂ 2xy^2 ÃÂ 4y^3
2. Make sure the terms are in a standard order. See example 4 for
terms that do not appear to have a standard order.
3. The first and last terms go in their traditional boxes.
4. The two middle terms become the ÃÂKey NumbersÃÂ
5. The top term is not necessary, but you can put the two middle terms
here so that the students see that this is really no different than a
trinomial with 2 middle terms.
6. If the checking box does not work in both directions, then it
cannot be factored into 2 binomials.
________________________________________
| Factor Table | 6x^2y ÃÂ 2xy^2 |
------------------------------------------------------
| 3x^2 | 2y | 6x^2y |
------------------------------------------------------
| x | ÃÂ 2y^2 | ÃÂ 2xy^2 |
------------------------------------------------------
| 3x^3 | ÃÂ 4y^3 | ÃÂ 12x^3y^3 |
------------------------------------------------------
Thus: 3x^3 + 6x^2y ÃÂ 2xy^2 ÃÂ 4y^3 = (3x^2 ÃÂ 2y^2 )( x + 2y)
Example #3
Sometimes it is difficult to decide on a standard order and the
students just work with the 4 terms until the ÃÂChecking BoxÃÂ works.
They begin to see the pattern quickly that the product of the two
outer terms must equal the product of the two inner terms.
Factor: 2ab ÃÂ 3cd ÃÂ bc + 6ad
Change the order so that the ÃÂChecking BoxÃÂ will work.
New order: 2ab ÃÂ bc + 6ad ÃÂ 3cd
_________________________________
| Factor Table | ÃÂ bc + 6ad |
---------------------------------
| b | ÃÂ c | ÃÂ bc |
---------------------------------
| 2a | 3d | + 6ad |
---------------------------------
| 2ab | ÃÂ 3cd | ÃÂ 6abcd |
---------------------------------
Factors: 2ab ÃÂ bc + 6ad ÃÂ 3cd = (b + 3d) (2a ÃÂ c)
CONCLUSION:
The thing that I like most about the factor table is that it gives the
student something to write immediately. Many students will just sit
and stare at the problem and hope the factors will miraculously pop
out. If they donÃÂt see the factors immediately, all I have to do is
say ÃÂMake a factor tableÃÂ and they get it quickly. I have not solved
the problem of removing the GCF first, but the factor table is so easy
that most students can still factor it with the GCF in place. (Maybe
they will notice it later.) This is one of the first things that I
teach all my students (from algebra 2 to calculus). Many of them tell
me that they could never factor until they used the factor table. It
makes factoring fun and easy. I know that factoring is not considered
an ÃÂinÃÂ topic with people who want to do it all with their graphing
calculator, but I still feel that factoring is a proper and necessary
skill for people who want to be well rounded in mathematics.
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