Cinderology
Sunday
May162010

## FactorTable.Com-- Factoring Trinomials

This unique, powerful Factor Table presented here was created last Friday, on May 14, 2010 or, perhaps, merely rediscovered. Nevertheless, we believe you'll find this concise format to be the most useful way to explore a myriad of potential trinomial factors.

If you know of a better way to organize your thoughts on trinomial factoring, please challenge us. We believe this is as good as it gets! Please share your comment below and alerts us to any known prior use.

ax2 + bx + c = 0

The Factor Table leverages what you know -- signs and the magnitude of the "bx" term -- to quickly illuminate what you don't know, rather than mindlessly exploring each and every option.

The essence of trinomial factoring is to reverse FOIL multiplication and discover the root factors. Our unique Factor Table organizes the prime factors in a concise format that allows you to then mentally explore all relevant possibilities by focusing on the "bx" value of the trinomial.

For example, our concise Factor Table for 6p2 +5p -6 is presented below. Just how this works will be explained as you read through the examples that follow. We could easily create a Factor Table with a spreadsheet and isolate the correct factor set without any mental effort! But that would defeat the purpose of algebra; to train your mind to think in logical, rule-based steps. Besides, it's quicker this way.

So keep this in mind as you learn factoring; we don't need these answers. We have whole warehouses full of "answers." What we need are trained minds; minds that can, by extension, solve real world problems using the same logical skills.

To train your mind, don't skip steps; don't let study partners think for you; and don't gloss over robust solutions and settle for mere answers. Rather, engage in deliberate practice until you have mastered the process. You are forging mental tools that, once mastered, will server to attack and solve problems for a lifetime.

The Art of Trinomial Factoring

The art of factoring trinomials begins with observing that they come in five basic formats:

• ax2 + bx + c: Signs-- both products are positive
• ax2 + bx - c: Signs-- smaller product is negative
• ax2 - bx - c: Signs-- larger product is negative
• ax2 - bx + c; Signs-- both products are negative
• ax2 - c;  Signs-- products are equal with opposite signs

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UPDATE-- For trinomials with "-C" value, the assumed sign assignment failed on occasion. We figured it out.

While most trinomials fall in line, a few defy the sign assignment assumed from the sign of the "BX" factor. One such example is 6x +x -15, which factors into (3x +5)(2x-3). Using our original logic, we would have predicted that the lower row should be negative, as the "BX" factor is positive and, thus, the larger products are positive. True enough, but the top row contains the larger numbers-- not necessarily the larger products.

The magnitude of the "BX" factors are the same regardless of sign assignment, so it was tempting to determine magnitude first and then place the sign. If that helps simply the process for you, please do so. Yet, I believe you will find the vast majority of the time the sign assignment is correct; larger numbers most often do yield the larger products. Further, you need to subtract the two products anyway, i.e. "add" a positive and negative product, so having one assigned negative is helpful. Do let us know what you think.

To resolve this occasional sign flip, we re-worded the sign assignment as tentative. Then, should you discover the "BX" magnitude with the wrong sign, just flip it. In the example here. you would have uncovered 3X(3) + 2X(-5)  for a -1X result -- the right magnitude but wrong sign. Just flip it and you're done, i.e. use 3X(-3) + 2X(5)  for a +1X result yielding (3X+5)(2X -3). Always double check your results by recombining the trinomial after factoring. In our example, we recombine for 6X +X -15, so this works.

Note-- If you didn't quite follow this discussion, no worries. Do a few examples below then revisit it. It will become clear.

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The Factor Table is set up with the larger "ax2" factors on top which forces the larger products on the top line. Signs are tentatively applied to the "c" factors on the top line and/or second line, depending on the format of the trinomial. Finally, once the correct "bx" magnitude is discovered, the sign is confirmed or, if necessary, flipped.

Here's how we create a Factor Table using 6p2 +5p -6 as an example:

• List Unique "a" Factors -- List unique factors for “a” in descending order, just once (not in reverse order), with the largest products on the top line. Each includes the variable and is separated by a vertical line or box. Here, in our example, 6p2 is the result of one of the following: 6p * p, 3p * 2p, 2p * 3p, or p * 6p. We place the larger factors on the top line, 6p and 3p, while ignoring the reverse order.

• List All "c" Factors-- Next, list all possible factors of “c” in descending order and then reverse order, so that each is listed twice (unless equal). Here' what we have so far. The larger "a" factors on the top, left followed by the full list of unsigned "c" factors: • Tentatively apply Signs -- Finally, we use the value of "bx" to tentatively apply signs to the larger numbers on the top line, the smaller numbers on the second line, or both. In our example, 6p2 +5p -6, the fact that we have a positive "bx" factor and a negative "c" factor, means that only one factor is negative, not both. But, which one? When the "bx" factor is positive, the larger product must be positive, so we tentatively assign the negative sign to the bottom row which, most often, combines for the right factors. • Together, this Factor Table gives us all useful, non-redundant combinations in a concise form. The larger products are always on the top row. The negative sign, when necessary, is tentatively assigned to the top, bottom, or both rows. The economy of the Factor Table comes from its unique two-line format and from eliminating all unnecessary combinations. We now have only the relevant candidates, so there will be no wasted mental effort.
• Mental Elimination. Finally we are ready to use mental math to discover the factors of 6p2 +5p -6. By design, all the "a" and "c" factors are already correct, i.e any of these factor combinations will create 6p2 +(b)p -6. It's the "b" factor that must be discovered among the candidates. We are looking the products that, when combined, will result is an (b)p" factor of +5p. Here's how it's done. • Mentally combine the factors as you go down the row beginning with first factor set, 6p/p and then test for +5p, the result we want: 36-1 is too high (Do this in your head: 6p * 6 = 36p, the first factor on the top row plus p * -1 = -p, the first factor on the bottom row. Ignore the "p" and just combine 36 + -1, or 36-1), then 18 - 2 is too high, then 12 - 3 is too high, and 6 -6 is too low, thus, we can eliminate 6p/p as a factor set.
• Next, do the same with the 3p/2p set:  18-2 is too high (from 3p * 6 on the top row plus 2p * -1 on the bottom row, ignoring the "p") but next 9-4 = +5, so we've found the right combination! The sign is correct, so we're done.
• Finally, keep in mind that we cross multiply factors -- the "OI" in FOIL -- to get trinomials, so after placing 3p and 2p in their customary first positions, we recombine the top row as outside "O" factors and the bottom row as inside "I" factors to get the correct combination:
• 6p2 +5p - 6 = (3p -2)(2p +3)

Let’s look at a variety of examples to see how the Factor Table is used to mentally uncover trinomial factors:

Using the Factor Table to Factor Trinomials

Example #1: x2 + 2x -15

Before you begin a trinomial factoring problem, be sure to check for scalers that can be factored out.

Had this been, x3 + 2x2 -15x, for instance, we would first extract an "x" to give us x(x2 + 2x -15), and then factor the rest. If the trinomial is set equal to zero, the variable "x" would be a solution, i.e. x = 0 is one solutions to x3 + 2x2 -15x = 0

In contrast, constant scalers do not create solutions and can simply be ignored. The trinomial 20x2 + 40x -300 can be reduced to 20(x2 + 2x -15), for the convenience of seeking smaller factors. Even more subtle, had the x2 term been negative, i.e. -x2 - 2x +15, we would first extract a "-1" to give us -1(x2 + 2x -15), an expression with both positive "x" factors, i.e. -x2 - 2x +15 = -1(x - 3)(x + 5)

Keep in mind that scaling constants like the "-1" have no impact when solving for "x." For example, if we have (x - 3) = 0, we'd find that -1(x - 3) = 0 yields exactly the same answer, i.e. the solution lies in x = 3 either way, so the -1, like any other scaling constant, can simply be ignored (see Example #7 below). Back to the current example which will show how a Factor Table works.

Example #1: x2 + 2x -15

Step 1) List factors : Since a=1, the Factor Table is set up with the “c” factors of 15, 15 * 1, 5 * 3, 3 * 5, and 1 * 15: Step 2) Set sign: Given b = -2, the smaller product must be is negative so we assign the second line negative. Why? The convention is to set the top line to hold the largest numbers. Here, a = 1 so this may not be clear. When a > 1 we will generate the larger products on the top line since the larger "a" factors will be placed there, when applicable (see later examples). Step 3) Mental Survey: Now we are ready to mentally go through the line-up looking for the which set of “c” factors will result in a +2x. We reject 15x -1x as too large but see that the 5,-3 set does this nicely, as 5x - 3x = +2x.

Step 4) Cross factor and confirm: Set the factors using the top line ax, (here a=1) with the second line “c” factor of -3; then the second line ax (here a=1) with the top line “c” factor of +5. Here this step is trivial as both "a" factors are 1, but the need to cross factor will become evident in more difficult problems.

This is necessitated because we aligned the "a" factors with the "c" factors in the Factor Table for our visual convenience, while the actual factors are crossed when written down, the "OI" in FOIL. The factors confirm the original equation using the FOIL method, so we have it right:

(x - 3) (x + 5 ) = x2 + 2x -15

Step 5) Check if either factor can be further reduced: No, so we’re done.

Example #2: p2 -2p -35

Step 1) List factors : Since a=1, the Factor Table  is set up with the “c” factors of 35, 35 * 1, 7 * 5, 5 * 7, 1 * 35: Step 2) Set sign; Given that larger product must be negative to end up with b = -2, we set the top line negative this time: Step 3) Mental Survey: Mentally go through the line-up looking for the which set of “c” factors will result in a -2p deficit. The -7, 5 set does the job nicely.

Step 4) Cross factor and confirm; Set the factors using the top line "a" factor with the second line “c” factor; then the second line "a" factor with the top line “c” factor.

(x - 3) (x + 5 ) = p2 -2p -35

Step 5) Check if either factor can be further reduced: No, so we’re done.

Example #3: 3p2 -25p + 16

Step 1) List factors: Since 3 is prime, 3 * 1 is the only "a" factor. The Factor Table is set up with the “c” factors of 16 * 1, 8 * 2, 4 * 4, 4 * 4, 2 * 8, and 1 * 16. Since 4 * 4 is redundant, we can ignore the reverse of this combination. Step 2) Set signs; Both lines are negative to get a positive “c” and a negative difference. Step 3) Survey Line-up: Mentally go through the line-up looking for the which set of products will result in a -25p deficit. Since no combination works, -48p -p, -24p -2p, -12p - 4p, -6p - 8p, or -3p - 16p, this trinomial cannot be reduced, i.e. it’s already "prime."

Example #4: 6p2 +5p -6

This is a good example of a trinomial that is difficult to do in your head alone.

Step 1) List factors: Since a = 6, the “a” factors are 6 * 1 and 3 * 2. Remember, we will exhaust all possibilities of the “c” factors, so we need not exhaust these. The Factor Table is then set up with the “c” factors of 6 * 1, 3 * 2, 2 * 3, and 1 * 6. Step 2) Set signs; Given that larger product must be positive to end up with b = +5, we set the bottom line negative this time: Step 3) Survey Line-up: Mentally go through the line up looking for the which set of products will result in a +5p, first using the 6p and p. None of these factors work; 36p - p, 18p -2p, 12p - 3p, or 6p - 6p, so the first set can be eliminated. Next, survey the 3p and 2p products; 18p - 2p, 9p - 4p, 6p - 6p, and 3p - 12p. You'll note that the second product works nicely yielding, 3p(3) + 2p(-2) = 9p - 4p = 5p

Step 4) Cross factor and Confirm; Set the factors using the top line 3p with the second line “c” factor of -2; then the second line 2p with the top line “c” factor of 3.

(3p - 2) (2p + 3) = 6p2 +5p -6

Step 5) Check if either factor can be further reduced: No, so we’re done.

Example #5;  Z4 -16

This example will demonstrate the need for Step 5

Step 1) List factors: Step 2) Set signs: Step 3) Survey line-up; Here we knew the only way to eliminate the “bx” term is to use the +/- the square root, 4 and -4

Step 4) Cross factor and Confirm; Again trivial as the terms are reversible.

(z2 - 4)(z2 + 4) = z4 - 16

Step 5) Check if either factor can be further reduced. Here (z2 - 4) is also reducible:

(z - 2)(z + 2)(z2 +4)

Example #6: 30x2 - 25x -30

This example highlights the concise format of the Factor Table that would, otherwise, take a half a page to explore all possibilities! In addition, you will see that you can skip over factors whenever the products are so large they are obviously out of bounds.

Step 1) List factors: Since a = 30, the “a” factors are 30 * 1, 15 * 2, 10 * 3, and 6 * 5. Remember, we will exhaust all possibilities of the “c” factors, so we just need the unique ones. The Factor Table is then set up with the “c” factors of 30 * 1, 15 * 2, 10 * 3, 6 * 5, 5 * 6, 3 * 10, 2 * 15, and 1 * 30. Step 2) Set signs; Given that the larger product must be negative to achieve -25x, we set the top line negative: Step 3) Mental Survey: Mentally go through the line up looking for the which set of products will result in a -25x, first using the 30x and x combination. You can skip over many of the factors as the resulting products are > 100. Looking from 30x * -3, we get -80x (-90x + 10x), -45x, and 0x, so we can eliminate 30x as a factor.

Next up, look over the 15x skipping down to 15x * -5, as the products of those that precede -5  are over 100. From there, we get -75x +12x = -63x but then finde -45x +20x = -25x and we’re done! The “c” factor set of -3, 10 works nicely in combination with “a” factors 15x, 2x.

Step 4) Cross factor and Confirm; Set the factors using the top line 15x with the second line “c” factor of +10; then the second line 2x with the top line “c” factor of -3.

(15x + 10) (2x - 3) = 30x2 -25x -30

Step 5) Check if either factor can be further reduced: No, so we’re done.

Example #7: x2/12 -2x/3 -4 = 0

Before we begin, we'll want to rewrite this expression into a more convenient form, so let's multiply by 12 to get rid of the fractions. Yes you can do this since 12 * 0 is still "0!" This becomes, x2 - 8x -48

Step 1) List factors : Since a=1, the Factor Table is set up with the “c” factors of 48, 48 * 1, 24 * 2, 16 * 3, 12 * 4, 8 * 6, 6 * 8, 4 * 12, 3 * 16, 2 * 24 and 1 * 48: Step 2) Set sign; Given that larger product must be negative to end up with b = -8, we set the top line negative: Step 3) Survey Line-up: Mentally go through the line-up looking for the which set of “c” factors will result in a -8x deficit. The -12, 4 set does the job nicely.

Step 4) Cross factor and confirm; Set the factors using the top line "a" factor with the second line “c” factor; then the second line "a" factor with the top line “c” factor.

(x + 4) (x - 12 ) = x2 -8x -48 = 0; Solutions x = -4, +12

Step 5) Check if either factor can be further reduced: No, so we’re done.

But, now let's go back to the original fractional expression and see if these solutions really work!

Let x = -4 where x2/12 -2x/3 -4 = 0

(-4)2/12 -2(-4)/3 -4 = 0

16/12 +8/3 = 4?

(1 + 4/12) + (2 + 2/4) = 4

(1 + 1/3) + (2 + 2/3) =4 Yes!

Let x = 12 where x2/12 -2x/3 -4 = 0

(12)2/12 -2(12)/3 -4 = 0

12 - 24/3 -4 =

12 -8 -4 = 0 Yes!