Bam!!!
Mr. Tarrou.
In this video we are going to continue our discussion of factoring polynomials and deal
with specifically the difference of squares.
There are some patterns that you want to memorize when you are learning factoring because they
show up so often.
They follow a special pattern that you don't need to... you know... actually really work
your way through the problem and take up extra time.
If you know the difference of squares pattern, you can just look at a lot of problems and
just see the answer like that.
When you have a squared minus b squared, now that is equal to when you factor it a plus
b times a minus b.
Now you might notice that our exponents of 2 are simply becoming exponents of 1.
That is because, or how that is happening is you are square rooting the first term to
get 'a', and you are square rooting the second term to get 'b'.
So if I just take this, you know, if I just take the square root of a squared.
Square rooting a squared number will undo... it is the inverse math operation of squaring
a number.
That is just going to equal 'a'.
That is where that is coming from.
So a is going to be the square root of the first term and b is going to be the square
root of the second term.
Now also, before I start these examples, if you want to verify that your polynomial in
factored form is actually indeed correct all you have to do is
distribute them together..
or FOIL it together.
I don't use this idea of FOIL, but your teacher probably does.
So you know, is (a+b) times (a-b) the same as a^2-b^2?
Well 'a' times 'a' is a squared.
When you multiply like bases you add those exponents.
'a' times negative 'b' is negative ab.
Positive 'b' times positive 'a' is positive ab.
And 'b' times negative 'b' is negative b squared.
Now when you are dealing with, you know, this special pattern where it is a difference of
squares where your two binomials that I am multiplying back together to check to make
sure that it works...
When your two binomial have exactly the same first and second term but the signs are different
(conjugate pairs) what happens through the distribution process or the FOIL process what
would have normally created your middle term... because normally when you multiply two binomials
you get a trinomial or three terms...
they will cancel out.
This gives you that difference of, excuse me, a difference of...
This minus sign is that negative sign... a difference of squares.
Your middle term cancels out.
Ok, so enough of the intro.
I have four examples and then I have a fifth one coming, sort of like a bonus extra credit
question.
A little bit harder than you might find in your textbook.
I am not sure.
But we have a difference of perfect squares.
Now I know this is a perfect square because of two and I know that 9 is a perfect square
because of course the square root of 9 is 3... or negative 3.
If this where a 7 you can write the square root of seven, but it is not identified a
perfect square.
You cannot square root that number and get a whole number answer.
So I am looking for perfect squares like 4 which is 2 times 2, 9 which is 3 times 3,
16 which is 4 times 4, and so on.
Now if I look at this I can sort of... you know... envision square rooting that first
and last term and I can identify that 'a' is equal to x, and 'b' is equal to 3 then
fill in my pattern.
So x squared minus 9 is equal to (x+3) or 'a' plus b times (x-3) or 'a' minus 'b'.
That is a pretty simple one.
Here in the next one we have a leading coefficient that is not equal to one, and two of variables...
or two of terms... both of our terms have a variable in them with an even exponent.
You could the difference of square process too if your exponents are odd, but you do
not normally see that if you are in Algebra 1 or even an Algebra 2 book.
So I am going to ignore those and just deal with exponents which are even.
So...
Now I am out of the camera screen.
Let's just stand over here.
So 4 x squared minus 16 y squared.
A difference of squared.
Well the square root of that first term is 2x, so 'a' is going to be equal to 2x.
The square root of the second term...
I am doing that slowly... because I am kind of like, that is what you should be thinking
in your head.
The square root of the second term 16y^2, notice I did not put the negative sign in
it by the way, you cannot square root a negative number.
The square root of 16 is four, so b is equal to 4 and the square root of y squared is y.
So 'a' is 2x and 'b' is 4y for this example.
So 4x^2-16y^2 is equal to (2x+4y)... a plus b... times (2x-4y)... a minus b.
Again, this is a pattern where we are supposed to just to be following it.
Hopefully now you understand how it works.
But you could distribute this together to make sure that it comes out to the original
problem.
Not a bad idea, because how often do you get problems on a test that you can check before
you hand in the test.
So distribute that back together and make sure that it works.
Here we have x to the fourth minus y to the fourth.
Now this is also a perfect difference of squares.
But you see the exponents, they are not 2...they are 4.
Again, a little bit more of an advanced problem.
Well just a second ago I wrote the square root of x squared or maybe I wrote a squared
or something like that.
The square root of x squared though is equal to x to the first.
The square rooting process, you know, will take your exponent and divide it by 2.
I don't want to get too complicated if you have not studied rational exponents.
But the square root, the index of that radical, whatever that is... that is how you are going
to change the exponents.
We are taking the square root so we are going to take all those exponents and divide them
by two.
I am pausing because I want to talk about cube roots and fourth roots and so on, but
we are doing a difference of squares, so I just want to focus on the square root.
It is going to going to take the exponents and divide them by two.
That is how you go from 2 to 1.
So if that where x to the fourth, the square root of x to the fourth is x squared.
The square root of x to the fifth, we are not going to see this in this video, but that
would be x to the 5/2.
Just going to throw that in there in case you have dealt with rational exponents where
the top number is the power and the bottom number is the index of the root.
Ok, so what does that mean?
Two terms, minus sign in the middle, so this is a difference of squares.
That means that I can square the first term, square root the second term, and identify
'a' as equal to x squared and 'b' as being y squared and walk through the difference
of squares pattern.
So this becomes x... a plus b, so x squared plus y squared.
That is the 'a' plus 'b' part.
This times x squared minus y squared.
Again the (a+b)(a-b).
Ok, well...
We are getting into factoring more and you don't always...
you are not always done after one step of factoring.
With this problem I once again have another difference of squares set up.
Now this is a good point to stop and also say, we can factor a difference of squares
but we cannot factor a sum of squares.
There is no such thing as..
Well there is such a thing, it is right there... sum of squares... but you cannot factor it.
So after factoring the original problem of x^4-y^4, we have something in factored form
but there is once again something in there that can be factored further.
So this is once again a difference of squares.
Our final answer is going to be x...
Let me just box this in because this is just scratch work.
We have x squared plus y squared times.
Now here we are going to factor that difference of squares again.
We are going to have x plus y times x minus y.
Our last example just looks like a binomial.
The first step of factoring should not be to look for special cases or if you factoring
a quadratic trinomial, that is a second degree trinomial or polynomial...trinomial, the first
step of factoring is to find anything in common that you can pull out of all of the terms.
Well both of these terms have a factor of 2, they both have x, and they both have a
y.
So I can factor a monomial out of these two terms.
We are going to take a 2x... they both 2 and 32 have a factor of 2... both of these terms
have an x and we are going to take out that lowest...
Let me get a different color here.
We are going to take out the lowest exponent of one.
Both of these term have a y.
We are going to take out that lowest exponent again of 1 and factor that out.
Now I just finished a whole video of factoring out monomials and I show the division step
where you pull that GCF out.
Now I am just going to do that in my head.
2 divided by 2 is equal to 1 which I am not going to write, but it is one.
x to the third divided by x to the first, if I have three x's and I am taking one of
them out leaves me with two of them left or x squared.
Minus because we are dividing by a positive number.
Minus 32 divided by 2 is 16.
I have one x and I am taking the only x out that this term has.
This second term has 3 y's and I am taking one of them out.
That is going to leave me with two y's.
Ahh..
Ok, so I had just a binomial but after factoring out the 2xy I get a difference of squares
again.
So this is a multi-step problem when we are factoring here with this one.
Let me get my blue chalk.
We will square root the first term...
Just a second.
Sorry for the interruption.
So we are going to square root the first term, square root the last term, and identify 'a'
as being equal to x, and 'b' as being the square root of 16y^2.
Well the square root of 16 is 4 and the square root of y squared is y.
So I have taken out my 2xy factoring out a monomial.
But I have that binomial left which is a difference of squares.
My final answer is going to be 2xy times... ok, let's see here.
'a' is x and 'b' is 4y.
It is going to be a plus b or (x+4y) times (x-4y).
I have one more example for you that sort of, it is an extra credit...
extra hard example.
Last example.
Now this one is a bit more complicated looking.
But these parentheses are locking these terms all together.
So this is something squared minus something squared.
So while it looks a lot more complicated, this is exactly the same as even the first
example.
We are going to identify this first term here 'a' as being...
Well what is inside?
What is being squared?
x-y, so 'a' is going to be x-y.
'b', well 'b' is in the second term and it is x+y.
So if you like, I can temporarily just say well that is 'a' and this is 'b'.
It is 'a' squared minus 'b' squared.
a^2-b^2 is a plus b times a minus b.
I am writing out the pattern out for you and we are just going to look at the formula because
this is kind of a complicated question.
Let's just say this is 'a' and this is 'b'.
This is something squared minus something squared.
That is a plus b times a minus b.
Ok, well these bases are a lot more complicated but the parenthesis are locking it all together.
'a' is x-y and a goes in the first two blanks.
'b' is x+y.
Now because this one is a bit more complicated than all of the other problems, we are going
to go ahead and simplify what is left.
Here we have x...
These parenthesis I have put in as placeholders.
But there is no coefficient, there is no exponent, there is no division, so it really is not
doing much.
It is really not doing anything except for highlight where 'a' was and 'b' was.
So x plus x is 2x, and negative y plus a positive y...well when I try to put those two like
terms together, negative one and positive one cancel out.
Negative one plus one is zero.
That is going to be just 2x.
This times, over here...
I am going to remind you that we do have a negative in between those.
Maybe even just show a little scratch work.
We have x-y, negative one times x is negative x, and negative one times y is negative y.
Ok, well what do we have?
We have x minus x which is equal to zero.
We have negative y and a negative y.
Well minus one, or negative one minus another negative one.. not minus a negative one...
but subtracting one is negative two, so -2y.
I could not add these factors.
I could not add these terms....they are not terms because there is not an addition or
subtraction sign in the middle.
But I can multiply them.
2 times -2 is -4xy.
That is your extra credit bonus.
I am Mr. Tarrou.
BAM!!!
Do do your homework:D