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Archive for the ‘Factoring’ Category

Factoring Trick: How to Flawlessly Factor any “Difference of Two Squares” Binomial

If you’re staring at two terms you need to factor, but feel like a deer looking at the headlights of an oncoming semi, here’s a way to leap to safety!

It’s called the “Difference of Two Squares” trick.

High-Octane Boost for Math

It requires four simple steps.

  1. Figure out if each of the terms is a “perfect square.”
  2. If so, take the square root of each term.
  3. Put each square root in its proper place inside two (    ).
  4. Put a + sign inside the first (   ), and put a – sign inside the second (   ).

Let’s do an easy example. Suppose the terms you’re looking at are these:
x^2  – 9

Let’s go through the 4 steps together.

  1. Figure out if each term is a “perfect square.”

    So, what does it mean for a number or term to be a “perfect square”?  It means that you get the number or term by multiplying a number or term by itself. For example, 16 is a perfect square because you can get 16 by multiplying 4 by itself:  4 x 4 = 16.

    So when we look at our two terms, x^2 and 9, we notice that both
    are perfect squares.
    9 is just 3 times 3.
    And in the same way, x^2 is just x times x.

  2.  Take the square root of each term.
    The square root of x^2 is just x.
    And the square root of 9 is just 3.

  3. Put each square root in the proper place inside two sets of (    ).
    We put the square root of the term that was positive first, and the square root of the term that was negative second.Since the x^2 was the positive term, we put its square root, x, first inside each
    (   ).  So far, that gives us:  (x    ) (x     )

    Since the 9 was the negative term because it had the negative sign in front of it: – 9, we put its square root, 3, second inside each (   ). So our (   )s now look like this:  (x   3) (x   3)

  4. Finally, we just need to put in signs that connect the terms inside
    the (    )s.

    That’s easy. We put a + sign inside one (    ), and we put a – sign
    inside the other (    ).
    I prefer to put the + inside the first (   ), but it really doesn’t matter.The final factored form, then, looks like this:  (x + 3) (x – 3)
    That’s all there is to it.

Now try these problems for practice.

           a)  x^2 – 16
           b)  x^2 – 100
           c)   x^2 – 121
           d)   x^4 –  16x^2
           e)   49x^8 – 144y^12


           a)   x^2 – 16   =  (x + 4) (x – 4)
           b)  x^2 – 100  = (x + 10) (x – 10)
           c)   x^2 – 121  = (x + 11) ( x – 11)
           d)   x^4 –  16x^2  = (x^2 + 4x) (x^2 – 4x)
           e)   49x^8 – 144y^12  = (7x^4 + 12y^6)(7x^4 – 12y^6)
















“Algebra Survival” Program, v. 2.0, has just arrived!

The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.


Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.








How to Factor Trinomials with Understanding!

This video shows the fastest and easiest way I know of for factoring quadratic trinomials. Give it a watch and see if you agree.

How to Factor the Simplest Kind of Quadratic Trinomials (a = +1)

Yep, factoring quadratic trinomials is a key skill for Algebra 1. And the process can seem intimidating, especially at first.

But it’s actually surprisingly easy if taught in a certain way. And of course, that’s what I’m going to do here … teach it in the easiest and fastest way possible.

Believe it or not, there’s a reason teachers make you factor trinomials. They may not have told you yet, but they do this so you can solve equations with quadratic trinomials. Once you can factor one of these little beasts, solving an equation that contains one becomes amazingly simple. But without the ability to factor the trinomial, solving it is much more difficult.

You’ll notice that this video starts with four preliminary concepts. These are pretty simple concepts, and for most of you these will feel like review. But make sure you know all of those concepts before you go on, especially the concept of absolute value.

With these preliminaries “under your belt,” factoring trinomials will be rather easy.

To put this video into perspective, it shows how to factor two of the four kinds of quadratic trinomials, those with the pattern of + + + and + – +. After this video, I will post another that shows how to factor quadratic trinomials with the patterns of + + – and + – –.

Also, my first two videos on factoring trinomials are for trinomials whose a-value = + 1. There’s a different, more complicated process for factoring quadratic trinomials whose a-value is not = + 1.  I’ll go over that in a few later videos.

In any case, this will get you started in a way that shouldn’t feel too painful. Follow along and good luck.





How to Decrease Algebraic Mistakes – Part 6

This is Part 6 in my series for helping students make fewer mistakes in algebra.

In this post I show how — by using the double-slash notation — students can avoid mistakes when factoring by grouping.

No Mistakes

Let's Reduce Mistakes in Algebra!


How to factor out the GCF with stories

At various times when I tutor, I find myself explaining the same concept repeatedly over several weeks.

Recently it has been that way with — drumroll please … factoring out the GCF from polynomials.

One reason I’m getting so much “experience” with this is that many kids find this process very difficult. It’s not hard to see why. First of all, the process of finding the GCF is, in itself, somewhat tricky. Then too, factoring out the GCF from all terms in a polynomial is a multi-step process; students need to get each step right, and then they need to perform the steps in the correct order. If that alone were not enough to tax children’s minds, students also get confused by the difference between how to multiply pure numbers (constants and coefficients), and how to multiply variables. (more…)

How to tell if 4 Goes into a Number — Divisibility by 4

My last post offered a neat trick for seeing if 3 divides evenly into a number.

In this post, I’ll do the same thing for the number 4.

But my approach will be a bit different in this post. Instead of just presenting the “trick,” I will help us grasp the logic behind the trick by looking at two principles of divisibility. I’m doing this because learning the principles should boost your ability to work — or should I say, play? — with numbers.

First, a question: If a number divides evenly into one number, will it divide evenly into all multiples of that number? Example, given that 6 divides evenly into 30, will 6 divide evenly into the multiples of 30, such as 60, 90, 120, 150, etc. The answer is YES. This is a basic principle of divisibility, and we’ll call it the Divisibility Principle of Multiples, or just DPM, for short.

Second, related question:  if a number divides into two other numbers evenly, will it also divide evenly into the sum of those numbers? Check this out with an example, and see if it agree with your mathematical “common sense,” aka “number sense.”

4 goes into both 20 and 8, right? So does that mean that 4 goes into the sum of 20 and 8, namely 28? Well, yes, 4 does go into 28 evenly, seven times in fact.

Test one more example with  larger numbers. 9 goes into 90 and 36, right? So does that mean that 9 also must go into 90 + 36, which is 126? Yes again. This idea harmonizes with “number sense,” and it is in fact true. And we will use this soon. We’ll call this the Divisibility Principle of Sums, or just DPS.

To get started thinking about divisibility by 4, let’s consider one nice thing about 4:  it divides evenly into a number that ends in 0,  the number 20! This is helpful because in our base-10 number system, numbers that end in 0 are “friendly” — they fit into the system neatly.

Using DPM, then, since 4 goes into 20, it goes into all the multiples of 20:  20, 40, 60, 80, and  yes, 100! Why is this a big deal? Since 4 goes into 100, we can use DPM again to say that 4 goes into all multiples of 100:  200; 300; 400;  … 700; 1,300;  2,300, … we can even be certain that 4 goes into 6,235,700 since this is a multiple of 100 [100 x 62,357  =  6,235,700]

The implication of this is major:  if we want to figure out if 4 goes into any whole number, we can ignore all but the last two digits. In other words, to figure out if 4 goes into 5,296 we need only ask: does 4 go into 96. The reason is that we already know that 4 goes into 5,200, and using DPS, if 4 goes into both 5,200 and 96, we can be certain that 4 will go into 5,296.

So we now have the first part of our trick for 4:  To find out if 4 goes into any number, look only at the last two digits.

That’s a great start. But we can get even more precise.

First ask:  before 4 goes into 20, what other numbers does 4 divide into? Simple, 4 goes into 4, 8, 12, and 16.

DPS, we recall, says that  if a number, let’s call it n, goes into two other numbers — call them a and b — then n goes into their sum:   a + b.

We can use this idea right here. Since 4 divides into 20, and it also divides into 4, 8, 12 and 16, DPS guarantees that 4 also goes into the bold numbers below:
20 + 4 = 24
20 + 8 = 28
20 + 12 = 32
20 + 16 = 36

Big deal, you say, since you already knew this from the times tables.  True, but  going up one multiple of 20, you can start to see the power of this idea.

Since 4 divides into 40, and into 4, 8, 12 and 16, 4 also goes into the bold numbers:
40 + 4 = 44
40 + 8 = 48
40 + 12 = 52
40 + 16 = 56

Once again, since 4 divides into 60, and into 4, 8, 12 and 16, 4 also goes into:
60 + 4 = 64
60 + 8 = 68
60 + 12 = 72
60 + 16 = 76

Using the same pattern, we see that 4 goes into:  80, 84, 88, 92 and 96.

Great, you might say, this shows us a pattern, but not a “trick.”
Where is this long-promised trick?

What we need to realize is that the pattern leads to a trick.

For the trick, here’s what you do:

1st) Take the two digits at the end of any whole number.

2nd) Find the lesser but nearest multiple of 20, and subtract it from the two-digit number.

3rd) Look at the number you get by subtracting. If it’s a multiple of 4, then 4 DOES got into the original number. If it is NOT a multiple of 4, then 4 does NOT go into the original number.

Words, words, words, right? Let’s see some examples to give the words some life!

Does 4 divide into 58?

—  Nearest multiple of 20 to 58 is 40.
—  58 – 40 = 18
—  18 is NOT a multiple of 4, so 4 does NOT divide evenly into 58.


Does 4 divide into 376?

—  Focus on the last two digits:  76
—  Nearest multiple of 20 to 76 is 60.
—  76 – 60 = 16
—  16 IS a multiple of 4, so 4 DOES divide evenly into 376.

Does 4 divide into 57,794?

—  Focus on the last two digits:  94.
—  Nearest multiple of 20 to 94 is 80.
—  94 – 80 = 14
—  14 is NOT a multiple of 4, so 4 does NOT divide evenly into 57,794.

Make sense? If so, then you are ready to do some serious divisibility work with 4. Here are some practice problems, and their answers.

PROBLEMS:  Tell if 4 divides evenly into the following numbers.

a)   74
b)  92
c)   354
d)   768
e)  1,596
f)   3,390
g)  52,472
h)  831,062
i)  973,236
j)   17,531,958


a)   74:  74 – 60 = 14.  4  does NOT divide evenly into 74.
b)  92:  92 – 80 = 12.  4 DOES divide evenly into 92.
c)   354:  54 – 40 = 14.  4 does NOT divide evenly into 354.
d)   768:  68 – 60 = 8.  4 DOES divide evenly into 768.
e)  1,596:  96 – 80 = 16.  4 DOES divide evenly into 1,596.
f)   3,390:  90 – 80  = 10.  4 does NOT divide evenly into 3,390.
g)  52,472:  72 – 60 = 12.  4 DOES divide evenly into 52,472.
h)  831,062:  62 – 60 = 2.  4 does NOT divide evenly into 831, 062.
i)  973,236:  36 – 20 = 16.  4 DOES divide evenly into 973,236.
j)   17,531,958:  58 – 40  =  18.  4 does NOT divide evenly int0 7,531,958.