BAM!
Mr. Tarrou.
One more time today.
Like the shirt by the way?
Every Friday is a funny shirt whether you like them or not.
Ok, so we are going to do a little factoring review.
um...
Definitely a sore spot, even with my some of my PreCalculus kids.
So I have got a few examples up here about factoring.
Let's take a look at them.
Ok.
The very first thing that you should try when you have to factor is ask if there is something
in common that all of the terms can give up.
Can you just simply divide out a common factor?
And with this binomial here they both have a 3.
15 and 3 are both divisible by 3.
And both of these terms also have an x to give up.
How many x's can you take out?
This has two to give up and this term only has one to give up.
When you are factoring out variables, you can take out the lowest number of those common
variables.
So these both have a 3 and both of these terms have an x.
We are going to take 3x^2 and divide it by 3x, and 15x and divide that by 3x and factor
out... pull out.. or undistributed the 3x.
The common factor.
So we have 3x...
3 x squared divided by 3x is just x.
15 divided by 3 is 5.
And x divided by x is 1.
So 3x^2+15x equals 3x times x+5 when it is factored.
Ok, this one here my PreCalculus students have not even seen much of yet.
And these two terms also have common factor.
Those two terms both an x.
But you might be going, "yeah but those have negative exponents."
"That is weird."
"I don't get that."
Well, I am going to do this just like this did this simpler example over here.
Both of these terms have an x in them so I am going to pull an x out.
Now how many can I take out?
Well I just told you that when you are factoring out a common factor, when you are factoring
out a common x in this example, you take out the lowest exponent.
This has two to give up, but this only has one... the lowest exponent.
Well what is lower, the negative five or the negative three?
On the number line -5 is farther to the left.
Negative five is the smaller exponent.
So that is how many we are going to take out.
So we get x^-5 is going to come out of x^-5 over x^-5, I am dividing out, plus 2x^-3 divided
by x^-5.
And if your class is not doing ones like this with negative exponents, you can ignore this
example... or learn from it.
But it might be beyond the difficulty in your class.
x to the negative fifth...
anything divided by itself is equal to 1, so x^-5 divided by x^-5 is simply 1... plus
2 divided by 1 is 2... and What do you do when you divide like bases?
If you have been studying exponents recently, you know that when you divide like bases you
subtract the exponents.
So this is going to be x to the -3 minus -5.
Well, what is -3-(-5)?
When you minus a negative, that is the same as addition.
So negative three plus five is equal to two.
Beautiful.
That is factored.
And there you go.
You might have never seen factoring out a common factor when the exponents are negative.
But they work just like your simpler ones do that maybe your are more comfortable with.
Now when you have a trinomial and no common factors come out, then the only chance you
have or the only way you have of factoring it is to set up your two parenthesis and start
putting factors of the first term in the first two blanks and factors of the last two terms
in the last two blanks.
Well some students always struggle a little bit with factoring.
So I don't want to say that it is easy, but when your leading coefficient is one as far
as factoring is concerned those are the easier ones.
You just simply just want to ask yourself what factors of this last constant either
add or subtract, in this case subtract because it says minus, to the middle term?
Factors of 24 that minus to the value of, or give you, a negative two when you subtract
them.
We have 1 and 24, 2 and 12, 3 and 8, and you have 4 and 6.
So if I write x + 4 and x - 6, these factors of 24 will subtract...
4 minus 6 does give me negative 2.
And if you want to check that, this is the correct answer, but if you want to check it
which is always a very good idea, then you just start FOIL'ing this back out.
x times x is x^2.
x times -6 is negative 6x. 4 times x is positive 4x.
And positive 4 times negative 6 is negative 24.
There you go.
Look, -6x plus 4x is your -2x.
So this answer does check out.
Very nice.
Alright.
What if you have a leading coefficient of one still, but your last sign is positive.
Instead of saying factors that subtract to the middle term, you will say factors that
add to the middle term.
What factors of 15 add to 8?
There are not many to choose from.
1 and 15 is 16, and 3 and 5 is 8.
So this will be x+5 and x+3.
If your last sign is positive, your two signs in your factors are always going to be the
same.
What if I change that 8 to a negative 8?
Well I just said when your last sign is positive, these have to be the same.
They don't have to both be positive, the both have to be the same.
So what factors of 15 add to -8?
Well, -5 and -3.
So if you are watching this video, I did just change the question on you.
Please pay attention to that.
Ok.
So there, that is factored.
Now!
Now we get to the issue... maybe.
Hopefully not once this is done.
Now we are factoring a trinomial with a leading coefficient that is not equal to 1.
Back in my days when I first teaching and learning Algebra it was all trial and error,
and it sucked!
[laughing] Ok.
i hated it.
Now they are like... hey guys...
There is a way to factor these trinomials where it is not trial and error.
It is just a little bit of scratch work on the side.
If you have a trinomial and nothing comes out, you alway want to look to see if there
is something that divide out of all three of them first...
If you have a leading coefficient that is not one, here is what you want to do.
Take the first and last term and multiply them together.
This is scratch work on the side.
First and last term multiplied together.
3 times -20 is negative sixty.
Talk about trial and error.
There are kinds of combinations that you can come up with.
So what we want to do now, is what factors of -60... what factors add to the middle term
of -11.
Well, you have 1 and 60, 2 and 30, 3 and 20, 4 and 15.
So and it has to be negative, so the four or the fifteen... one of those has to be negative.
That is going to be, if I want to get -11, it is going to be -15 and positive 4.
Now what I am going to do is rewrite this to look different.
I am going to rewrite the middle term with these factors which I did not write down of
positive 4 and negative 15.
So this is going to be 3x^2 minus 15x plus 4x minus 20.
Again, these two middle terms still add up to negative eleven so all I have done is made
the expression look different.
The -15 and positive 4 came from factors of -60 that added to the middle term.
Now I am done with my scratch work.
Now I have four terms instead of three.
What do you do when there is four terms?
You do something called Factor by Grouping.
And when you factor by grouping you work with the first two terms and the last two terms
separately temporarily.
What comes out of the first two terms?
They both have a 3, 15 is divisible by 3.
And they both have an x.
Now what is 3x^2 divided by 3x?
3 divided by 3 is 1.
And x squared divided by x is x.
-15 divided by 3 is 5, negative 5... and the x's will cancel out.
Now I am going to take something out of the last two terms to make this factor of x-5
show up again.
What does 4 and 20 both have in common that I can divide out and give me x-5 when I am
done inside the parenthesis?
Positive four.
So I am going to divide out a positive 4.
4 divided by 4 is equal to 1.
20 divided by...
NEGATIVE 20 divided by 4 is negative five.
Now look, I have matching factors again.
So we have x-5 times 3x-4.
You can check it, but that is the right answer.
And that is factoring.
WHOOO!!!