Factoring Polynomials is easy, once you know how how...

Out of the entire method factoring polynomials gets the most "ooooh's and "aaaahs" even more than problem solving.

Algebra can strike fear into the toughest foot ball player, make grown women cry (and some men too) and the saddest part is THERE IS NO NEED FOR IT.

ALGEBRA IS EASY!!!

If presented correctly everyone and anyone can understand it. EASILY. I have been in room with 750 people and they all "got it!" Youngest to oldest. 5 year olds and seventy year olds alike.Read carefully te next few minutes you spend here could change EVERYTHING.

Before we start factroing plynomials, first lets back up, review the five concepts, talk about the words and what they mean and use some examples before we dive in to the algebra since for many you may have found your way here via search engine and you may have not been exposed to very much concept based manipulative mathematics before this.

Also forgive me in advance as I am going to rob many of you of the ah-ha experience that comes with this presentation unless you have a set of blocks handy and can resist looking ahead. For teachers and parents there is method here, and good reasons why we do the things we do.

Algebra has it's own math vocabulary words. Here we have two of them put together "factoring polynomials." In kid speak factoring means 'form a rectangle; count the sides'. This will make more sense in a moment.

Next we have a great math vocabulary word: polynomial. "Poly" from the greek meaning "many" or "more than one" and "nomial" meaning number.

If we look at the symbols or the blocks we can see that it's a number made up of more than one part.

Mononomial: 1, x, x² are mononomials, ONE number.

Bi-nomial: x + 1, x² + x, x² + 1

Tri-nomial: x² + 3x + 2 after this we just call them polynomials...but the term includes everything except mononomials...poly means more than one after all.

In a polynomial we don't know what "x" is but we do know the x in x² is the same number as the "x".

So putting all that together, we are going to make a rectangle and count the sides out of the many parts of the polynomial. Still not clear? Well lets take a look a "regular" number like 18.

We've played with 18 before. Look at all the ways we can reshape 18.

Here are our old friends the sixes. See how we can take 18 and turn it into a rectangle? One side is 3 the other side is 6.

These are two of the factors of 18, the factors are the sides. We've played with 18 and other numbers and we know we can form it into rectangles. Note the side is just 3 or 6 we only count the edges. This is important to emphasize. Just the sides, not the insides.


Here are our old friends the sixes. See how we can take 18 and turn it into a rectangle? One side is 3 the other side is 6.

These are two of the factors of 18, I used sixes both times even though any kid can tell you the first one should be made out of threes.

(3)(6) = 18 Three, six times is eighteen. (6)(3) = 18 Six, three times eighteen.

GO HERE for a discussion on Multiplicand and Multiplier. Now that we have some confidence built up and understand that all we are doing is building a rectangle and counting the sides, lets move on to a not so regular number like

x² + 3x + 2.

Factoring this polynomial means we need to get out the right blocks and form them into a rectangle. I always start this with a simple three period lesson so they know the names of the blocks...and then tell them to make it into a rectangle. This gives them pause often after they just heard and did the boy scout story. {link not built yet} Algebra fears run deep and mental blocks come up as soon as they see these symbols.

If you have base ten blocks or cut outs handy try it without looking below and give yourself the ah-ha experience. If you are teaching students for damh sure let them discover this for themselves don't do it for them! I put big pictures here so you will have to scroll down to see more.

Really resist the temptation and try it for yourself. If you are a student STOP! Don't go any further. Take these manipulatives out and try and make a rectangle with them. It's self correcting: you cannot fail. All the pieces must be together and you can't leave out the green ones or stack the blocks on top of each other. You may come up with several variations from the one I show you below...as will your students. They are all correct; the one below is easiest to count.

But a picture is worth 1000 words.

Examine this picture and look at the factors of each block. (Check out my totally awesome self portrait.)

Wait a second...what are the sides?

Factoring polynomials is as easy as 1, 2 3!

We can count the sides. Here, let Crewton Ramone show you. He's pointing to the across factor, which is (x + 2). You might have built it the other way, it doesn't matter. When you are factoring polynomials all that matters is that the symbols match the blocks for across and up. Encourage children to take their finger and sweep it along the across factor. This way they get more senses involved and it becomes that much easier. The fingers and the tongue are connected via neural pathways, get the kinesthetic learners even more involved, and help those who are "challenged" with dyslexia or learning disabilities understand better. Also works with adults.

Over then up! The UP is x + 1.

Some ask, "why isn't it 2x + 2?"

Because we are only counting the sides, NOT the INSIDE, remember? Just like with 18 and the sixes, one side was 3 the other 6 we didn't count inside. It's the same with factoring polynomials as it is with regular numbers.

Now even trying to make it look all complicated with lots of symbols and stuff...it's not. However this is a good lesson on focus. With all that stuff in there it might be hard to find what you're looking for. Some people may not "get it". Too busy. Too much to look at.

So lets put you in a situation where you cannot fail.

Cover one side with a piece of paper here I have a drawing but you should use the blocks all you can see is the across now which is (x + 2). It's not three we can't count them all at once because they are not same. The blue is x and the the two green ones are two. It's visually obvious.

We just factored the polynomial x² + 3x + 2.
and found the sides were
(x + 2)(x + 1).

x² + 3x + 2 = (x + 2)(x + 1).

The first part of the equation is the whole rectangle, the second part is the sides. Little kids find this almost redundant. In person this whole page about factoring polynomials takes five to ten minutes to present...soon there will a video on factoring polynomials...less than ten minutes clears up a lot of problems with factoring polynomials. Did we talk about FOIL? Did we have to go into the distributive theory of multiplication? NO. But we will, but not to start with and certainly not with the toddlers...we like this one because it puts the student in a position were they can not fail. This builds confidence which helps with all their math.

Right now we are done crawling...and we are about to take baby steps...this was just the first example...I'll give you several more and then you'll be able to factor ANY positive trinomial with one as the coefficient for x². Factoring polynomials is fun once you get the hang of it, 1st graders ask for it by name: "can we play algebra?!"

They think it's silly and hardly believe me when I tell them adults and high school kids think factoring polynomials is hard.

Here it is again but instead of factoring it, I made it into a division problem...which is a whole lot easier because you get the rectangle and one side.

I know what some of you are thinking...

"Big whoop, that one was easy...what about ones like

x² + 7x + 12 ??

and what about negative expressions?

One thing at a time. Go to factoring tri-nomials for more. {which STILL isn't built yet} Hopefully many of you can also see how cool this is...

Right now let's continue with the "next" one:

x² + 4x + 3

Now that you've built x² + 3x + 2 you need to do another one, sometimes this is so new to you that even when you see it you still have a hard time believing it can be this easy. All we had to do is count the sides...? No way.

Yeah way.

Cognitive dissonance can occur. SO quick try the next one. I'll even give you a rule, put one on the top and the rest on the side...

x² + 4x + 3



Go HOME from Factoring Polynomials

Go here for more examples of Factoring Trinomials.

My blog has lots of examples of algebra on it.

“I learned just enough in school to figure out that everything is not all there is to know.”

"The whole purpose of education is to turn mirrors into windows." ~Sydney J. Harris

"The simplest schoolboy is now familiar with truths for which Archimedes would have given his life." ~Ernest Renan, Souvenirs D'enfance Et De Jeunesse, 1883