While factoring polynomials is not
division, it is kind of the opposite of
multiplying. So, let's review multiplying
polynomials. Let's try and multiply this
expression. I'm going to distribute the
2x to both the 6x and the 5. 2x times 6x
is 12x squared then times the five and I
get 10x. Now I'm done with the 2x and I
have to do the 1. 1 times 6x is 6x and I'm
going to line him up with the 10x
because they're getting together later.
And 1 times 5 is 5; now, let's add like
terms and we get 12x squared plus 16x
plus 5
Now let's scan our answer and see where
each part of it came from, The 12x
squared I want you to note, the first
term really came from the first two
terms of our problem and the five the
last term came from the last two terms
of our problem. Now, the 16x, did he come
from nowhere?
I can't see, well actually, I want to say
he came from... a little bit corny, the
rainbow.
He came from a combination of the
outsides 2 times 5 and the inside. See
the rainbow? 10 plus 6.
Let's do another problem just so you can
see what's the same every time. I'm going
to distribute the X; that will be 3x
squared minus 7x. Now I'm done
with the x and I multiply the one.
That's going to be 1 times 3x. I'll put
him under his buddy and 1 times negative
7. Now adding like terms I get 3x squared
minus 4x minus seven. Once again, where
did this answer come from?
Let's look at each term. The 3x squared,
the first term, came from the first parts
didn't it?
The negative 7, the last term, came from
the last parts and the middle term came
from the rainbow.
That's a negative 7x and a plus 3x. If you
combine the two you get negative 4x. So
using that idea we're going to factor by
working backwards, backwards of
multiplication.
Ok. Let's try and come up with the answer
to the factoring of this problem.
Well, I'm going to have to make guesses.
Let's make a reasonable guess for these
two numbers. We know they have to
multiply the 12x squared. How about 6x and 2x
That's a reasonable guess and a guess for
these two they have to multiply to 5. It is
only a guess but I'll put one and five.
Now, am I correct? The rainbow has to give
me 16x. Let me check it. I'm going to do what
you call "check the rainbow".
That's a 30 and a 2 and that makes 32.
That's not it.
So I can admit I'm wrong and don't erase
this 1! I just try another.
How about another possibility for 12x
squared? 3 and 4 or 3x and 4x
and for 5, well, that's an easy one, 1 and 5.
Now, once again, let's check the rainbow
to get the middle term. I'm going to get
15x + 4x is 19x. This is a tough one.
The bad news is: it is a bunch of guesses.
The good news is you have to check it to
do it so you know when you're going to
be right. So, don't give up. Let's try
again. Another guess: 2x and 6x. I switch
the order; order counts and for 5, 1 and five
again. Let's see what we get for the
rainbow 10. 2 times 5 and 6, 1 times 6,
Oooh, that looks good and
we are absolutely positive that we're
correct because we check the answer so
that way, if you would, will work every
time and you can be guarantee that
you'll get a hundred. Let's try another.
Watch what we're doing here.
Make a reasonable guess for this one.
Well, there's only two possibilities to multiply
to 5x squared. 5x and x. Now, 6 or negative 6,
let's try a negative one and a positive
6. That's a reasonable guess. I'll check the
rainbow to see if I get the middle term.
Well, 5 times 6 doesn't look good, does it?
30 and minus 1; this one doesn't work but
once again I remind you: don't erase it
on your paper so you're not checking it
over and over again, the same one.
Okay, Let's try again. I'm going to try the
same 5x + 1x but for 6 this time let's
try two and three.
See if I'm going to get the middle term
by checking the rainbow. I get a negative
15 and a positive 2; now a negative 15
and a positive 2 give me a negative 13x.
I, wait a minute, I want a positive 13x so
I'm close.
So, what I'm going to do is only switch
the signs since I'm so close. So let's
use 5x and 1x and this time negative 2
and positive 3. We'll check the rainbow
and I'll get positive 15x and minus 2,
I got it! It really isn't genius; it's
just sticking with it until you get it.
Now they're all about the same but let's
try another. What's going to multiply 2x
squared?
Well, sometimes it's so easy it's hard.
It's going to be X and X; that's all it
could be. Now for 12
I hate 12. There's a bunch of numbers that
multiply to 12 but we'll try three and
four and you're going to see, you know
all right away I bet you know that
I'm wrong. Three and four
give me 7 and you're gonna get, you're
going to get good at this
you're going to be making the right
guess earlier because what's going to
happen here is you're going to be
reviewing your multiplication tables.
We'll try x and x again here and this
time for the 12
let's try six and two and that's going
to give me, for the rainbow a positive 2
and a positive 6 and I'm going to get
positive 8x. I'm very close. Now what
happens when I get the right number but
the wrong sign?
Well. Let's switch the signs.
Once again, X and X and this time
remember I want a positive 12. I can get
a positive 12 with two negatives, negative 6 and
negative 2 and if I check the rainbow
I'm going to get negative 2x and
negative 6x and I know I'm right.
Hopefully you got the idea. Something
that looks as different as this will
still work the same way. My guess for
this term is going to be XY and XY and my
guess for this term, I'll use eight and
one and I check the rainbow and I get 1
XY and 8xy. Annh I probably shouldn't
have guessed 8 and 1.
Let's try again XY and XY and I bet
you already know what to put here. Let's
put two and four and we'll get 4XY + 2XY
and there's your boy!
This one looks very different but we can
still do it the same way.
Remember I didn't say first second or
third. This is sometimes called the
difference of two squares but we don't
even need to know that. It will work the
same way but it could be written
x squared plus none x minus 16. The middle term
could be nothing.
Okay? But you're probably not going to
see it that way. As usual these two
numbers are going to give me the first
value and these two numbers are going to
give me the last Value. I didn't say third; I
said the last value. Now why do we know
we're wrong here? 2x and minus 8x give
me minus 6.
Well, we need the pair of second numbers
to add up to 0.
Well, so what we're going to have is X
and X in this case and what else
multiplies to 16? A square.
Well, a positive 4 and a negative 4
add up to zero and you've got your
answer.
Here's another difference of two squares.
It could be written
y squared plus none x minus 25; you won't see
it that way. It'll stick out like a sore
thumb because there's two expressions
or I should say two terms rather
than three. So to do this we'll just do
the same factoring method. Make our
guess and then what's your guess going
to be for the last two? Now you know what
you're doing. Minus five and a plus 5 and
there it is.
You're getting better at this, aren't you?
There you go. Now this one's a little bit
more difficult and
we're going to add a step for you so
that we can minimize the number of
guesses we make. Let's factor out the greatest common factor first. We know how
to do that.
I'm looking at all three terms and I see
that a five can come out. Let's take it
out; it'll make the problem easier. Remember
how to take out the greatest common
factor. Basically it's going to be three
little division problems. That's why we
learned this and 70 divided by 5 going to be much
easier to deal with as 14. Well no, I'm
not done yet.
Now we bring down our five and now we
break up the blue part here the way I
just showed you. Make your guesses. What are
you're going to guess here?
Well, let me check the rainbow. Ewww, I'm
off by a little bit. That's all right.
I'm very unlucky. I'll check the,
I'll switch the signs. Just switch the
signs in this case.
So we bring down the 5, the greatest
common factor; make your first guess;
make your last guess and don't forget
check the rainbow. Plus 2 and minus 7.
We got it. And there's your answer.
And that's the way to do it. Now taking out
this greatest common factor is going to
help you a lot.
Let's take a look at this one. The
greatest common factor of these three is
2x squared. We talked about this before.
Nobody likes that negative in front so
instead of using 2x squared let's take
out negative 2x squared and then we'll
be done with the negative for the rest
of the problem.
Well, let's see what happens. Do our three
division problems and we have a somewhat
easier problem. I'm not saying it's easy.
This is the big leagues now but at least
it's doable.
Now we still need to continue breaking up the
blue part, if you would, so I'll bring out
my greatest common factor and break up
the blue part. This one's not that bad
because what are the possibilities for
3x squared? And what are the possibilities
for 11?
Let's check it.
Negative 3 and positive 11. I'm still not
very good at this guessing part am I? But
I'm not going to give up. That's the key.
I'm not going to give up. All I'm going to do
is switch the signs.
I'll do it.
3x + X this time minus 11 and one and I
get a 3, a minus 11 and I know I'm right.
It doesn't get much harder than that.
How about that? One last problem.
Note that we're going to do the greatest
common factor first. Don't do it because
I; said so do it because it makes the
problem easier. Now the greatest common
factor of these two terms is 10x to the
third so let's take it out and do our
division. Division here.
Okay, now, fear not. Look at the blue
part. We still have to break up the blue
part. Don't even look at 10x to the
third anymore.
That's the difference of two squares. It
sticks out like a sore thumb.
Let's break it up.
Remember that we could write this like
this. We don't have to but we could which
means my guesses are 2x and 2x and +3 and
minus 3. See, we're going to get our plus 6
and minus 6 and there's your answer.
So don't forget to always at least try
to take out the greatest common factor
first.