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Posts tagged ‘factoring’

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

Answers:

           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)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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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.

 

 

 

 

Good Question


Isn’t it great when kids ask good questions?

(Rhetorical question, that, of course.)

I got a good question today, about factoring.

I was showing this student how to factor by taking out the GCG, and he asks me, “So what’s the difference between factoring and dividing?”

You see, we had been using dividing when factoring the GCF. For example, to factor an expression like 4x + 16, we divided both terms by 4 after seeing the 4 is the GCF. So in this boy’s mind, factoring seemed akin to dividing.

What I liked about the question is that it made me think … and clarify something.

I realized that when you factor, you do divide, but you do more than divide.

Essentially, when you factor, you use division to make rename an expression. 

In the example I gave, you equate 4x + 16 with its factored form, 4(x + 4)

When you divide, on the other hand, you are just doing a small piece of this.

You divide, for example, when you ask:  4x divided by 4 = what? Answer: x

You use that answer to lay out the factored version, but dividing is only a step.

So hooray for good questions and congratulations to those who ask and recognize them.  Good questions make the act of teaching come alive.