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Factoring
Posted:
Jul 30, 1998 1:11 PM


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 email 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 trialanderror 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 clearcut starting place and sequential steps and an immediate check on their work.
VERTICAL TEAMING (Prealgebra) The object in a prealgebra 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 prealgebra examples to get the students familiar with the operation of the table.)
PREALGEBRA 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 prealgebra 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.



