Really Important Factoring Question

I am not sure what you mean by the “hit or miss” method. After a while doing this, like any skill, people get to the point where they see, or think they see, the answer fairly quickly and they just write down what they think is right — is that what you mean?

My preferred method is intelligently guided trial and error and correction — more on that in a minute.

I strongly recommend *against* the super-clunky method of multiply these two numbers and try to find a sum and a couple of products — it takes a horribly long time, requires just as many guesses as my method, and is not logically clear at all.

**Intelligently guided trial and error and correction**:

Big picture: factoring is just “un-multiplying”. You are pulling an object apart into the pieces that made it up when they were multiplied. So **practice multiplication** of algebraic expressions until you are very good at it, because until you know multiplication you will find it very hard to undo.
ALWAYS check a factoring by multiplying back. If it doesn’t work, first review your multiplications, then re-check the factoring.

Step 1 — ALWAYS pull out *all* COMMON FACTORS FIRST.
This is very important because it makes the rest much easier, and in particular with trinomials it can cut the work by as much as eight times.

Step 2 — COUNT the terms and CLASSIFY —

2A. Two terms — either unfactorable,
— OR difference of two squares (common)
a2-b2 = (a+b)(a-b)
— OR more rarely, advanced courses, sum or difference of cubes
a3+b3 = (a+b)(a2-ab+b2)
a3-b3 = (a-b)(a2+ab+b2)
— common errors to note: i. Sum of squares a2+b2 is NOT factorable
in real numbers! ii. That second part in factoring the cubes is NOT the
perfect square, in fact it is usually (unless higher powers involved) NOT
factorable.

2B. Three terms — binomal system, see below

2C. Four or more terms, factor by grouping: most often two groups of two terms (very rarely one and three); pull out common factors in each group and then re-group.
Example:
3x2y + 6xy2 - 4x - 8y
= (3x2y + 6xy2) - (4x + 8y) [group, pull out common negative]
= 3xy(x + 2y) -4(x + 2y) [pull out common factors in each group]
Look! we have a common binomial factor!
= (3xy - 4)(x + 2y) [common factor and re-grouping]
Now multiply back and check it!!

3. Trinomial system:

Think how multiplication works
(x + 5)(x - 4) = x2 - 4x + 5x - 20 = x2 + x - 20
(2x + 3)(5x - 2) = 10x2 - 4x + 15x - 6 = 10x2 + 11x -6

Now, the first term in the trinomial is just the product of the first terms in each bracket: x times x is x2, 2x times 5x is 10x2. The last or third term in the trinomial is just the product of the last terms in each bracket, +5 times -4 is -20, and +3 times -2 is -6.
The *middle* term in the trinomial is the problem because it is the sum of the two first-last products. We leave it and work it out later.

OK, to pull a trinomial apart, you look for ways to factor the first term, ways to factor the last term, and then you test until you get the right middle. This is logical and organized, and with a touch of practice it is very quick

Examples:
factor 3x2 + 11x + 6
Factors of the first term — since 3 is prime, only one choice, 3x times x
Factors of the second term — 6 = 1 x 6 or 6 x 1 or 2 x 3 or 3 x 2 (order matters, which way you pair it up with the x’s)
Signs are all positive
Possiblilies are
(3x + 1)(x + 6) = 3x2 + 18x + x + 6 = 3x2 + 19x + 6 Nope
(3x + 6)(x + 1) = 3x2 + 3x + 6x + 6 = 3x2 + 9x + 6 Nope
(3x + 2)(x + 3) = 3x2 + 9x + 2x + 6 = 3x2 + 11x + 6 GOT IT!
Well, we don’t even need to check the last one but it would be
(3x + 3)(x + 2) = 3x2 + 6x + 3x + 6 = 3x2 + 9x + 2

Note that I got this in three tries, and with a few further simplifications (below) I could have had it in one or at most two. This is *quick*, much quicker than the multiply and add and factor and split up again idea, and it makes sense.

Ex: factor x2 -x -1
Factors of x2 are only x times x
factors of -1 are only +1 times -1
Only possiblilty is (x+1)(x-1) = x2 - x + x -1 = x2 + 0 - 1 = x2 - 1
our old friend the difference of two squares
So x2 -x -1 is UNfactorable.

**Second advantage of this approach is that you can prove absolutely that something is unfactorable, no guesswork.

To speed this up even further:
Well, you DID remove all the common factors first, right? If not, you are making much more work that you have to. Since you *know* you don’t have common factors, throw out any test multiplications that have common factors.
Ex: Looking above at 3x2 + 11x + 6, we can immediately omit the patterns with 3x + 3 and 3x + 6 because we know there is no common 3.
So all we have to test are
(3x + 1)(x + 6) = 3x2 + 18x + x + 6 = 3x2 + 19x + 6 Nope
(3x + 2)(x + 3) = 3x2 + 9x + 2x + 6 = 3x2 + 11x + 6 GOT IT!

And even further:
No hard and fast rule, but extreme values are less common than middling values. So if you have for example 24, the factors 1 x 24 or 24 x 1 are always *possible* and never forget to list them, but 4 x 6 or 3 x 8 are a lot more likely. So start by testing the middling factors and then work out to the extremes.
Using this policy, 3x2 + 11x + 6 = (3x + 2)(x + 3)
= 3x2 + 9x + 2x + 6 = 3x2 + 11x + 6
Got it first try!

Example: Here’s a difficult-looking one, but we can crack it
factor 16x3 - 28x2 - 30x
(1) common factors? Yes!

16x3 - 28x2 - 30x = 2x(8x2 -14x -15)

(2) we have a trinomial, so break it sustematically

(3) OK, now that first and last thing:
Factors of 8x2: 8x times x, 4x times 2x, 2x times 4x, x times 8x
We only need to do the first two; we’ll switch orders with the last term and get all the combinations.
Factors of 15: 15 times 1, 5 times 3, 3 times 5, 1 times 15
Signs: for -15, we need a positive times a negative.
Start testing the middling values first (the factoring 8 times 1, 15 times 1, possible, try after the others fail)
(4x + 5)(2x - 3), (4x + 3)(2x - 5) most likely, no common factors

(4x + 5)(2x - 3) = 8x2 - 12x + 10x - 15 = 8x2 - 2x - 15 Nope
GOT IT!!
(and in only two tries)

[Note — if you did the signs reversesd you’d get
(4x - 3)(2x + 5) = 8x2 + 20x - 6x - 15 = 8x2 + 14x - 15
so if this happens, change the signs and test again]

Now, finish the job, making sure you have ALL the parts

16x3 - 28x2 - 30x = 2x(8x2 -14x -15)
= 2x(4x + 3)(2x - 5)

Keep It Simple, keep it logical, and practice. It works.

I’ll explain the “not recommended” multiply & get the sum — because it takes time but it works very well for a fair number of my students… but first to emphasize that the “difference of squares” thing is incredibly common in some form or other. (Also *very* common is to have something like a^4-81 which would factor out to a^2 + 9 and a^2 - 9… and then you have another difference of squares to factor out.)

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm has a graphically pleasing explanation of the trial & error approach that also (like Victoria’s explanation) helps structure it so it’s fewer trials and fewer errors

http://mcraefamily.com/MathHelp/factoring3c.htm explains the “sum & factor” approach, clunky as it is :) (It’s interesting — one or two sites even said that “the three ways” to factor were trial & error, completing the square, and the quadratic formula…. tho’ “sum and product” is really a systematized trial and error…)

And there is always the quadratic formula, though! If you want a truly brute force appraoch you plug your a b and c into it, and then you’ll get your two “roots” for the plus & minus…. I can explain better if you’d like…

What I am wondering is this…

Can one just use the quadratic formula for factoring?

Or

Does one have to do the hit or miss trial and error thing too?

Can one get away with (I hate to phrase it like this) never learning the hit or miss method with the guessing and all of that and just do the quadratic formula instead?

BTW

HAPPY NEW YEAR EVERYONE!!!

Can you use only the quadratic formula for factoring?

Well, how many years do you want to spend on fiddling calculations and then re-doing and re-re-re-doing those fiddling calculations when you get them wrong?

If you practice factoring, you can break most problems in a couple of minutes, often in seconds. The quadratic formula is going to take five to ten minutes per problem.
Then if the quadratic formula gives you a fraction or a decimal you have to do even more work to re-create the factors out of it.
You *must* check back the quadratic formula answers by multiplying anyway, in case you made a mistake.

Now there are two issues here — real factoring problems or fraction simplifications etc where you need to see the factors; and fill-in-the-blanks get the answers to this quadratic questions. For a real factoring problem the quadratic system is a drag. Possible but slow and error-prone. For a get-the-answers nobody cares how type of test, use whatever works.

If your only goal is to get through this present exam and if you have extended time on the exam, use what you can.
BUT when a question tells you to use a certain method, such as factoring, and you are marked on your work, know that you will lose a lot of marks by not following the method required — the goal was to teach a skill and if you refuse to use that skill you won’t get the marks.

If your goal is to go further and take another course, it is going to be assumed that you can use the basic skills in this course, so dont skip them; they do *not* go away if you ignore them.

You really do want to learn the process — which is *not* really hit or miss — for figuring out the possible easy answers.
Problems that are testing you on some other skill often include an easy – well, easy if you know the easy way — piece of factoring before you get to that next skill.
Let me ask you this, so I can better understand how your mind works with math (or doesn’t, as the case may be :-)) — Can you reduce fractions to their lowest terms?

SO how’s it going :-) Parkland College folks are back, but the Big University isn’t…
(and that’s my post above; they switched my ‘puter and it didn’t automatically log in ;)

well…

I worked through the entire gosh darned, acursed “Forgotten Algebra” book that is put out by Barrons. I cannot reccomend that book enough if you are slow like me and if you need nice instructions on how your ti 83 plus calculator ought to work along with the mathematical topic at hand. If I actually earn a passing mark in this class, I will find out who Mr. Barron is and marry him, I tell you!

And, I am in the process of completing my formal review that my instructor assigned on Tuesday; I figure I will be done with that by this time tommorow. I take math online and I first had to read over the schedule and all of that. The class is extra nice because our instructor informed us of a hyperlink that works with the text and it is like a video of an instructor…so if you have dyscalculia you can rewind the video, which is something you just plain cannot do in the class room setting! There is also a tutorial that shows the equation and all of the steps and even narrates it and everything! I honestly was never informed of such a thing previously at all, so I am still giddy about it.

The factoring is something, but I appreciate the hyperlinks of sue and all of your nice responses. What I find is that I can actually read the equation and then I am like really focused on taking the root of the two main numbers before anything. I am really obsessive with that and prime numbers, isn’t that weird? So, I am still reviewing, only this time for a grade and stuff because that is the first week of school type of thing. My instructor is to be grading all of the tests and quizzes by hand with some special device for on line courses, if any of you proper educators would like to know of it ask me and I will look it up in the e mail my professor sent me after I e mailed her about my ld.

I thank you all and peace!!

well…

I worked through the entire gosh darned, acursed “Forgotten Algebra” book that is put out by Barrons. I cannot reccomend that book enough if you are slow like me and if you need nice instructions on how your ti 83 plus calculator ought to work along with the mathematical topic at hand. If I actually earn a passing mark in this class, I will find out who Mr. Barron is and marry him, I tell you!

And, I am in the process of completing my formal review that my instructor assigned on Tuesday; I figure I will be done with that by this time tommorow. I take math online and I first had to read over the schedule and all of that. The class is extra nice because our instructor informed us of a hyperlink that works with the text and it is like a video of an instructor…so if you have dyscalculia you can rewind the video, which is something you just plain cannot do in the class room setting! There is also a tutorial that shows the equation and all of the steps and even narrates it and everything! I honestly was never informed of such a thing previously at all, so I am still giddy about it.

The factoring is something, but I appreciate the hyperlinks of sue and all of your nice responses. What I find is that I can actually read the equation and then I am like really focused on taking the root of the two main numbers before anything. I am really obsessive with that and prime numbers, isn’t that weird? So, I am still reviewing, only this time for a grade and stuff because that is the first week of school type of thing. My instructor is to be grading all of the tests and quizzes by hand with some special device for on line courses, if any of you proper educators would like to know of it ask me and I will look it up in the e mail my professor sent me after I e mailed her about my ld.

I thank you all and peace!!