BAM!!!
Mr. Tarrou.
In this video we are going to be factoring out a monomial from a polynomial.
What we are going to do is we are going to find the GCF.
That means that we are going to look at the coefficients, all of the numbers and...you
know...look at those numbers and find the biggest number that divides into all of the
coefficients evenly.
That is going to be numerically the GCF.
Then if any of the terms share a common variable, we are going to use the lowest exponent that
we can find for that common variable.
Like if all of your terms have an x in them, maybe one has x^5 and one has x^2, we will
be able to take out the lowest exponent or x squared.
Once we decide and work out what the GCF is we are going to divide that out.
Your teacher may be calling it un-distributing that GCF.
You should be learning this just after you learned the distributive property or multiplying
polynomials together.
So we are doing the opposite, so we are going to un-distribute that GCF.
Then if your problem is a bit more advanced and they will be in following videos, you
are going to want to continue to look at what you have left and factor.
You can take more than one step to get your polynomial completely factored.
With this video we are just doing the monomials.
We are just pulling those out or un-distributing those so we are not going to have any additional
steps I believe in the 1, 2, 3, 4, 5 examples we are about to work through.
So let's get to those examples.
BAM!!!
Ok, first example.
We have 2 x cubed plus 10.
Now these two terms both have a factor of 2.
I am not going to write all of these examples like this, I don't think.
But we can write 2 x cubed plus 2 times 5.
I am taking that constant and showing you that both the first term has a factor of 2
and so does the second.
So if both of your terms have a common factor, a GCF, then that is what you want to take
out.
Now this has three x's to give up, but the second term does not have any.
So I cannot take any x's out.
They don't...these two terms do not have any common variables.
But I can take out that value of two.
So I identified this GCF.
Ok, identify it and then un-distribute it.
That means we are not going to multiply it through the parenthesis, we are going to divide
it out.
You will start to mentally do this in your head, but when you are first learning this
concept you might want to show that division.
We are going to write 2x^3 divided by what we feel our GCF is... what we can factor of
all of our terms in this polynomial...
So we are going to divide that by 2 and 10 divided by 2.
So we are taking that GCF of 2 out of those two terms.
This multiplication of course is the inverse operation of this division.
So these math operations are just balancing out.
I am just making the polynomial look different, it is not changing its value.
So 2 divided by 2 is 1, and we have 2 times x cubed, and 10 divided by 2 is equal to 5.
Viola.
Why are we learning factoring?
The first purpose of factoring in your algebra 1, or algebra 2 course, or whatever you happen
to be in is that it helps you solve equations...finding out where they cross the x axis... find the
roots, the zeros, the solutions, whatever your textbook is calling it.
It will have other purposes later on in mathematics.
But initially after this you are going to be learning or using this skill for solving
equations.
Here we have 6 x squared plus 15 x.
Now both of these terms have a common variable so we are going to pull an x out and we are
going to factor out...
We are going to look through this and ask where is the lowest exponent.
That lowest exponent on the variable is 1.
So we are going to be able to take out...
Where is my colored chalk?
We are going to be able to take out an x.
Now between the 6 and the 15, do they share a common factor?
If they do, is it the largest common factor that they both share.
Like if your two numbers are 12 and 24, 12 and 24 both have a factor of 2, they both
have a factor of 3, they both have a factor of 4, but your biggest factor...
your biggest common factor...
6 as well... but your biggest common factor is 12.
You want to identify what can divide evenly into both of those coefficients and use the
largest number that you can factor out of both.
6 and 15 share a common factor of 3 and that is it.
We are going to take that three and divide it out.
We have 6x^2 divided by 3x plus 15x divided by 3x.
Again once you start to comfortably and consistently right you can skip this step, maybe if your
teacher allows you, and start to do more of this in your head.
We have 3x, then 6 divided by 3 is 2 and x squared divided by x....
Now I have not done....
I have done a video for properties of exponents dealing multiplication of the power rule but
i haven't done a quotient rule yet.
So hopefully you have in your class some experience here with this.
But we have x squares, that means there's two x's on the top and we have one x in the
denominator.
Well on of those x's...
Let's rewrite this since I know i have a cover.
Sorry for the interuption.
Let's go and rewrite that x squared is x times x cause it's kinda bothering me just a little
bit that i haven't covered the quotient rule or the division rule for exponents.
If you write that x is x squared, then you can see that our... let me get a different
color here... x divided by x is gonna be equal to 1 so that two on the top and you're taking
the one out with the division.
It's only gonna leave you with one x left over.
So we have 6 divided by 3 is 2 and x squared divided by x is one x.
And 15 divided by 3 is 5.
And I'm starting to write the x but they're going to cancel out.
Anything divided by itself is equal to 1.
So, there we go!
If you wanna check this answer like you know maybe you haven't dealt with dividing exponents
very much yet, and you wanna check your work, you can always just take this and multiply
it back through and make sure that it matches.
3 times 2 is equal to 6, x times x is equal to x squared.
Remember when you multiply those like bases you add the exponents.
And there's little unseen exponents of one there.
And 3x time 5 is equal to 15x.
And indeed that is what we started off with.
Now, we're gonna step up the difficulty a little bit in these examples as we go through
them as always.
9 and 21.
I'm gonna go ahead and take a negative number out.
Doesn't really maybe matter, depends on what your teacher wants.
I'm gonna go ahead and take a negative out just to change the sign of that first term.
And between 9 and 21 the largest common factor is 3.
So we're gonna take a negative 3.
Now in this one I have more than just one variable so I need to look at the x's and
y's separately.
Both of these terms have an x so I can take that x out.
If only the first one had an x and the second one did not, then I would be moving just to
the... move on to y variable to see if I can factor that out of... or divide that out of
each of those terms.
But this has 5 to give up and this has 2.
I can't take more than any one of the terms has.
So again I'm gonna take that lowest number of x squared.
And this has 4 y's to give up and this has 7 so we're gonna take 4 of those out.
And again we're un-distributing or dividing out these GCF's.
So we got negative 9 x to the fifth, y to the fourth over 3... negative 3 x squared
y to the fourth plus 21 x squared y to the seventh.
And we're going to divide that by negative 3 x squared y to the fourth.
Okie Okie.
So we have negative 3 x squared, y to the fourth.
Negative 9 divided by negative 3 is 3.
We have 5 x's and we're taking 2 of those out.
So that's going to leave us with 3.
Again 2 of these x's in the denominator is going to cancel out with 2 of those x's in
the numerator and there's 5 up there.
So you're gonna be left with 3.
And anything divided by itself is equal to one.
So this is just gonna cancel out and we have 3 x cubed.
Now there is a plus sign there and I almost just plus.
But as you work through your problems you should be doing the math as you go and just
don't start putting down you signs all over the place.
I mean, we're dealing with positives and negatives, you wanna make sure those signs are right.
So before you just write without thinking, positive 21 divided by negative 3 is negative
7.
Now you can write a plus negative 2, but I like to just use a single sign.
So 21 divided by negative 3 is negative 7.
X to the second divided by x to the seconds is gonna be.. that's just gonna cancel out
and I'm gonna go ahead and cross these out as well.
And this has 7 y's and you're taking 4 of them out.
That's gonna leave you with 3.
And that problem is factored.
I've got 2 more examples to finish this video.
Alright!
5 x to the fourth minus 7 x cubed plus 18 x.
Now this example I don't have any common factors between 5 and 7.
They're both prime.
And neither one of those numbers go into 18.
I just don't want you to think that you can always factor out a number and a variable.
So 5, 7 and 18 do not share any common factors.
However, all 3 terms have an x in them.
So I can take an x out of all 3.
This has 4 to give up, 3 to give up, and only.. whoops.. only one to give up.
So that's what we're gonna take out.
So we have x.
Now I'm just gonna take this out, kinda sip this step of writing the division.
If I have 5 x to the fourth and I'm taking that one out, or one of those x's out through
division, that's going to leave me with 3 x's.
So this is gonna be 5 x to the third minus 7 x squared.
I have 3 in here and I'm taking one of them out through division.
And then plus 18.
Well this term only has one x and I'm taking it out, I'm dividing it out and it's now in
front of the parenthesis so that's it.
If you want to check it, just take that x and distribute it back through.
Sorry for the interruption.
School's out and my contract is over and they're still on the intercom. 4 x cubed plus 12 x
to the negative 2 power.
This is a bit of a bonus question.
If you're just in say algebra 1, maybe not even your algebra 2 book might not have this.
I'm not sure.
But I'm having this because I teach precalculus and I'm getting my... brushing off my skills
for calculus and post some videos for that as well.
And there's a lot of instances where you need factor a monomial but some of your exponents
are negative.
You might not even learn that yet about negative exponents and how to deal with them.
Probably is coming up soon if you haven't.
I'm gonna work through this anyways, just call it an extra credit problem.
These two terms, these leading coefficients both have a factor of 2 but they also have
a factor of 4.
Of course that's bigger.
So we're looking for the GCF.
Both of these terms also have a variable of x, so we can take that common x out that's
in both of those terms.
But it's a little bit weird that one of those exponents are negative.
Remember, in your directions I said for common variables you wanna take out the lowest exponent,
and negative 2 is less than negative 3.
So we're gonna factor out that negative 4 x to the negative 2.
So that's gonna be 4 x cubed divided by 4 x to the negative 2 plus 12 x to the negative
second power over 4 x to the negative second power.
Okay well that's gonna be 4 x to the negative 2.
Anything divided by itself is one, so those four's are going to cancel out.
When you divide like bases, again...
I am getting ahead of myself a little bit because I have not taught the video...or taught
you how to... the quotient or the division property of exponents.
But, when you divide like bases you are supposed to subtract those exponents.
It is the opposite of when you multiply like bases you add those exponents.
So x to the third divided by x to the negative two is x to the 3 minus negative two.
I am doing that subtraction down as we normally would and we are subtracting that negative
exponent.
Now subtracting a negative is basically those two negatives are going to multiply together
and give addition.
This plus 12 divided by 4 is 3.
Again anything divided by itself is equal to 1...so that is just 3.
We have 4x^-2.
3 minus -2 is the same as 3 plus 2, so that is going to be 3+2 is 5.
We have x to the fifth plus 3.
Now, again I don't if you have studied negative exponents yet or not.
If you have not, it is probably coming up soon.
We don't like to leave negative exponents in our final answer.
So this base that has the negative exponent, we are going to move that to the denominator
of this...
Now there is no fraction.
It just looks like a single line of algebra here.
But we are going to create a fraction by taking this base with the negative exponent and floating
it down to the bottom.
Because I want to change that negative exponent to a positive exponent.
I explained in another video exactly why that happens, or give a couple of examples of why
or how that happens.
But for now we are just going to say the 4 and the (x^5+3) stays in the numerator and
the x to the negative two...
We are going to drop that x to the denominator so its exponent becomes positive.
So that is our answer and our final example.
I am Mr. Tarrou.
BAM!!!
Go do your homework!