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Hello, and welcome the Bay College's intermediate algebra
online lectures.
I'm Jim Helmer, and today we're going to be talking
about section 5.2, which deals with the addition and
subtraction of rational expressions.
This video is actually broken into two parts.
The first part, we're going to look at dealing with rational
expressions that have the same denominator, and then we're
going to look at least common denominators, a tool that we
use when we don't have the same denominator.
So that's the extent of this video.
In the next part of 5.2, we'll actually look at some examples
of rational expressions with noncommon denominators.
We'll have to find the like denominators before we can do
any addition or subtraction operations.
But first of all, let's start with reviewing what we should
already know.
When dealing with addition or subtraction of fractions we
have to have a common denominator.
If the denominators are the same, we simply need to add
the numerators.
It's saying we have 7 parts out of 12 and 11 parts of 12,
a total of-- if we add them together, we get 18
parts out of 12.
So a little bit more than one, right?
Let's leave it as an improper fraction.
But one thing we must always do, which we've been told to
do ever since we learned how to work with fractions, is to
always reduce your fractions.
Well, reducing fractions is something that we do in
rational expressions all the time.
We essentially factor and reduce.
That is the key to working with rational expressions.
Let's look at this and say, well, what are
their common factors?
Well, if we factor it down, I know that 12 is 6 times 2, and
I know 18 is 6 times 3.
So we have this common factor.
Any number divided by itself is 1.
1 times a quantity doesn't change it.
So we can think of it as canceling.
It's not really just going away, it's reducing to 1.
1 times 3/2 is 3/2, a reduced fraction.
So our answer is 3/2.
The same thing applies to subtraction.
Obviously, we know the difference between addition
and subtraction.
If they have the same denominator, we can subtract
their numerators 29 minus 22 is 7 over that common
denominator of 35.
7/35.
Again, we have to reduce, and we could do that by factoring.
35 is 7 times 5.
We can reduce this to 1.
Now, don't think of it as canceling, because it is
reducing to 1.
There is still something in this numerator, even though we
crossed it out, there still remains that factor of 1,
because 1 is a factor of every number.
So 1 over 5.
So our 7/35 reduces to 1/5.
This concept is exactly what we're going to use when we
deal with rational expressions.
Let's look at this first example here.
We have x plus 1 divided by 7 plus 6 divided by 7, or we can
say x plus 1/7 plus 6/7.
Because the denominators are the same, we can add our
rational expression.
Well, adding rational expressions is merely
combining like terms.
So if I'm adding these two numerators, I'm going to
combine like terms.
We can see 1 and 6 are just numbers.
They're like terms.
There's only this one x term.
So I have x plus 1 plus 6 is plus 7.
Now, one thing I want to point out just to make sure you
understand the concept, you cannot reduce 7 over 7 in this
problem, because we can only cancel factors.
These are terms.
In order to cancel it, I'd have to cancel the entire
quantity x plus 7, and I can't do that, because I don't have
that common factor.
It must be a factor before we can cancel it.
This is as far as we can go.
This is in lowest, or simplest, terms.
Let's look at the next example.
It gets a little bit more complicated because we have
more terms here.
Well, it's the same concept.
We have the same denominator, so we need to add the
numerators.
Well, adding the numerators, we have these two binomials.
We have to combine like terms.
2x plus 3x is 5 x's.
1 plus positive 5 is 6.
And in the denominator, we have x minus 3.
That doesn't change.
That's common to both of them.
A common mistake some students might do initially is also add
your denominators.
Do not do that.
It's just when they're common, you add the numerators.
Because this and this are both divided by the same thing.
So when we combine them, they're both divided by the
same thing.
Let's look at another example here.
Here we just have the quantity 1 over 3y minus 1.
And we have 12y minus 5 over 3y minus 1.
We want to add these.
So I'm just going to combine the numerators.
12y.
No other y term.
And negative 5 plus 1.
Be careful of your signs.
This is a negative 5 plus 1, which gives me negative 4 over
3y minus 1.
Now, in this example, 5 and 6 have no common factors, so I
can't factor anything out.
Because these are terms, I can't reduce them, I can't
cancel my x's, I can't do anything with that.
That was as far as we could go.
But if we look at this one, I see 12y and 4.
They have a common factor.
So maybe I can factor it down and reduce.
It's all about factoring and reducing to get it into its
simplest form.
Here, I could factor a 4 and that's going to leave me with
3y minus 1.
Now, this quantity, I can reduce it because this
difference here is a factor.
It's in parentheses here.
It's being multiplied by something.
The definition of a factor is something being multiplied.
So 3y minus 1 is divided by 3y minus 1.
Anything divided by itself is 1, so this can reduce to 1.
1 times this 4 I factored out is just 4.
We could also think of it as 4 over 1, but 4 divided
by 1 is still 4.
So we can factor and reduce.
Always make sure that your rational expressions are in
simplest form, and that's our goal.
Let's look at the next example.
What if we have subtraction?
Well, subtraction isn't necessarily that much
different, we just have to be careful of signs.
The most common error made in any math class, regardless of
what level you're at, are sign errors.
So we're going to be a little bit careful when we see
subtraction or we see negatives.
Even your instructors make sign errors.
We're not immune to them.
If we look at this example, we identify yup, they have the
same denominator, so I can just do what I need
to do on the top.
Well, here it's subtraction.
I have y plus 1 minus this numerator of 3.
So I have y minus no other y term, and I have 1 minus 3.
Since my numbers are like terms, 1 minus
3 is negative 2.
Both of these are divided by y plus 2.
They have the same denominator.
The denominator doesn't change in addition or subtraction.
Now, this is in simplest form.
We cannot cancel anything.
I can't cancel y's.
I can't cancel 2's.
We can't do that, because these are terms, not factors.
So that's reduced.
If we look at this next example, be very
careful with the sign.
There's two ways to approach a problem like subtraction of
this nature.
And one thing is, we can distribute this negative
through the numerator, because it says minus
whenever that numerator.
is, minus that quantity.
We could distribute that and change this sign to addition.
So a little bit of sign changes.
The other way to think about it is just take it
one piece at a time.
We've identified that they have the same denominator, so
I don't have to change that.
It is what it is.
Now here, I just take it-- let's deal with the x's.
5x minus 2x.
Make sure you keep track of that negative.
5x minus 2x would be 3x.
Then I have 4 minus a positive 7.
Well, 4 minus a positive 7 is the same as 4 minus 7.
4 minus 7 is negative 3.
Now, just like we did in this example
here, factor and reduce.
Let's check.
Is there anything I could do with this?
Well, I could factor out a 3.
I see that common factor in these two terms.
Once I factor it out, I've turned this into a factor,
because it's being multiplied by something else.
The quantity x minus 1 times 3.
Well, now this factor can cancel this entire factor.
Notice, both terms need to be canceled as a factor.
So this can cancels that, or reduce it to 1.
1 times 3, the only the quantity left, or 3 over 1, no
matter how you look at it, is just 3.
All right, so we got 3 there.
Let's look at another example.
This one's a little bit more math intensive, let's say,
because what we have to do is--
we recognize that we have the same denominator, so we can
just subtract the numerators.
Again, it's subtraction.
So maybe we think about it as distributing that negative,
making this negative 2x and positive 7 and putting an
addition sign there, or we can just keep track of it.
Let's just combine our like terms.
I'm going to work downwards here, because this one's going
to be a little bit more work to simplify.
What's my numerator going to be?
3x minus 2x is just an x.
Negative 1 minus a negative 7.
Be careful of those signs, right?
Negative 1 minus negative, well, that's addition, the
opposite of subtraction, minus a negative 7.
So negative 1 plus 7.
Negative 1 plus 7 is a positive 6.
Now, if we look at this, well, I can't factor the top, but is
it possible to factor the bottom?
Now, hopefully your factoring skills are relatively strong.
That is one tool to be very successful in any algebra
class is to have a very good foundation of factoring.
So maybe you can look at this and say, what are the factors
of a negative 6 that sum to 5?
Well, if you can recognize those, you're ready to factor.
x times x is x squared, and the factors of negative 6 that
sum to 5 are positive 6 and negative 1.
6 times negative 1 is negative 6.
6x minus 1x is a positive 5x.
If we were FOIL that out, we'd get right back to this.
One way to check our work.
Now, so I've dealt with this.
I've just factored it.
I've changed it a little bit.
This is what the denominator is.
I can see x plus 6 as a whole factor including both terms is
the same as this factor down here, x plus 6.
So now I can cancel the entire factor x plus 6.
What's on top is on bottom.
Anything divided by itself is 1.
So we reduce it to 1, there is still conceptually a 1 on top.
So it's 1 over x minus 1.
So you can see how this example was a little bit
different than the previous ones, because we weren't just
checking the top to factor, we recognize this and say, hey,
this factors, as well.
So maybe we can find some factor that's going to reduce,
as we did here.
So that's adding and subtracting rational
expressions that have the same denominator.
Well, now we're going to look at just some examples that
have unlike denominators.
We have to make them common denominators.
If you want to move the camera for the whole board here.
If we look at this example, we're going to look at
something that we probably have worked with in the past.
It's just the adding of fractions, but they don't have
the same denominator.
One of our rules is we must have the same denominator in
order to add or subtract any value.
Well, what we can do is we can use our factoring skills.
I can say, well, what is 15?
It's the factors 3 and 5.
Well, what is 6?
We can also factor that.
That is the factors of 2 and 3.
Now, to determine the least common denominator, the
definition of a least common denominator is essentially
having all the factors to their highest powers.
Well, if I look at this, I have a factor of 3.
Well, that's common to both, so I'm going to definitely
need that to get my LCD.
So I need a factor of 3.
This one has a factor of 5, this one does not.
I need a factor of 5 for both of them.
So we got to have that factor of 5.
And then I look at this one.
This one has a 2, but this one doesn't.
It's going to need a 2, so I must have a
factor of 2 in my LCD.
Now, once I've determined what all these factors are, I can
just multiply them together.
3 times 5 is 15.
15 times 2 is 30.
So my LCD it's 30.
Now, if you think about it, if I know what the factors are
here, I can go back to this and say, this needs a factor
of 2 in order to be 30.
15 times 2 is 30.
So I'm going to give it a factor of 2.
But I can't just change the fraction.
I end up changing its value by doing this.
Essentially, what I need to do is multiply by 1, and one way
to do that is to multiply by 2 over 2.
This value is 1.
1 times anything doesn't change the value, but in this
case, it'll change its appearance.
2 times 7 is 14.
I'm just going to write it over here.
2 times 15 is my common denominator of 30.
Now, let's look at this fraction here.
It has the factors of 2 and 3.
It needs a factor of 5 in order to be my least common
denominator.
So I'm going to give it that factor of 5, but what I do to
the bottom, I must do to the top so that I'm essentially
multiplying by 1.
5 times 5 is 25, and that 5 times 6 gives me that common
denominator of 30.
So we determine what that LCD is and we say what factors
does this one need to meet that requirement of an LCD.
And now we can just add it.
14 and 25 is 39 over that common denominator of 30.
And of course, now we can reduce.
I can factor this and say, you know what?
This is 3 and 13.
This is 3 and 10.
And now we just reduce, like we did in
the previous examples.
A 3 reduces the 3.
3 over 3 is 1.
1 times 13/10 is 13/10.
So we simplified that.
We're going to do the exact same thing when it comes to
our rational expressions in the next video, but for now,
let's just practice on determining what that LCD is.
In this example, I have 17x over 4y to the
fifth, and 2 over 8y.
We're not adding.
We're not subtracting.
We're just going to look at what would be the common
denominator of these two different rational
expressions?
Well, if I look at this, 4 and 8, they have a
common factor of 4.
4, and this would be 2 times 4.
So this would need that factor of 2.
So coefficient-wise, what is my LCD?
Well, if we just look at the coefficient, I know 4, if I
give it a factor of 2, it would be 8, if I
multiply it by 2.
So 8 is the factor we want to achieve, just like the number
here, 30, was the number we wanted to achieve.
But let's look at the variable here, y.
When I had mentioned initially we want all the factors and
their highest power, well, if this has five factors of y, in
order for this to be common, it also needs
five factors of y.
So we need five factors of y, that
highest power right there.
This power's 1, that powers 5, I need five of them.
So if we look at this, this is my LCD.
I looked at my numbers and I looked at the variables that
lie in the denominator.
So we have 8y to the fifth.
If I was doing any math operation to this, I could
give this that factor of 2 and give this the factor that it
needs, four more y's.
And what I do the bottom, I do to the top.
But in this case, we were just looking for the LCD.
Let's do the same thing to this example here.
Well, this one, because we have terms, we want to factor
a little bit.
We want to find these factors so we can distinguish, what
are our unique factors?
What are their highest powers?
So if I factor 7x minus 14 to 7 and the quantity x minus 2,
just factor out a 7 from both terms.
I can see well, I have a factor of 7 and I have a
factor of x minus 2.
This one is kindly already factored for
us, x minus 2 squared.
So if I want to find the LCD to this one,
what are the factors?
Well, I have a factor of 7, so I need at least
one factor of 7.
Here I have x minus 2.
Here I also have x minus 2, but it's squared.
This says I need the highest power of these factors.
Well, I have two of these factors, so I still need two
of those factors.
This is the LCD to this example.
If I had to change this denominator to be this LCD, I
just have to give it that additional
factor of x minus 2.
But if I do it to the bottom, I have to do it to the top.
Essentially, multiply by 1.
Let's look at this example.
It's our last example for just finding the LCD.
And then we'll look at one example where we actually use
the concept and apply it to an equation.
Now here, I look at this and I recognize, this
is a special product.
It's the difference of squares.
x squared is a perfect square, and 25 is a perfect square,
and we have a difference between them.
Differences of squares factor to x plus that number and x
minus that number.
Difference of squares gives me two factors
of different signs.
This one here, if I look at ut, I say, you know what?
They both have an x squared in common.
I can factor that out.
And they're also divisible by 3.
So this has 3x squared times x minus 5.
So what is the LCD for this one?
Well, let's look at what all the factors are.
I have a factor of 3 as my number.
That's the only individual coefficient out there.
And I have two factors of just x by itself, so I need that.
Now, I have x minus 5 in both of them.
It's highest power is just 1, so I need an x minus 5.
And then over here, I have this unique
factor of x plus 5.
You can see how sometimes in these rational expressions
these denominators can get pretty big.
But the key to these are, once we've factored and determine
what the LCD is, just leave it in this form.
Carry this through any operation you're doing.
Leave it in a fully factored form.
When we get to the end, it's already factored and we can
reduce anything that's common.
So our LCD is 3x squared times quantity x minus 5 times
quantity x plus 5.
A very large LCD, but it is the LCD for that example.
If we move over here, we're going to see a proportion.
Essentially, what a proportion is is a
fraction equal to a fraction.
Now, what we want to do here is just like we did there.
We want to factor.
We want to say, what are the factors?
So if I look at this one here, I could say, OK, well, these
two terms have a factor of 3.
So I'm going to factor that out and I get x
plus 2 as my factor.
3 times quantity x plus 2.
Here, this thing is already factored down for us.
I can't factor it any further, so I'm going to look
at, what is my LCD?
Well, my LCD, I have a factor of 3 in both of just one time,
so I need that factor.
I have a factor of x plus 2 in both of them.
This one has a factor of y.
So my LCD needs that factor of y.
So these are the unique factors and they're
multiplicity, or the number of times they repeat.
They each repeated just once, at least in one of our
denominators.
Now, if we look at this, that is already my LCD.
So if I look at this, what is this one missing in order to
be just like that one?
We're missing a y.
So if I want to find this value, let's make this a
common denominator.
Multiply this by y.
Essentially, I'd have 3y times that quantity, my LCD.
But what I do to the bottom, I also have to do to the top.
So I have y times this quantity.
This is what makes it equal to x.
By finding the LCD, I say, I know what factor to give this
to make this proportion a true statement.
Now, there's two different ways you could write it.
You could distribute that y, but I would
recommend you don't.
Because if we want to do further math down the road, if
it's in a fully factored form it's going to be easier to
reduce or multiply or divide or whatever we
have to do with it.
So this question mark is the quantity 4x
plus 1 times the y.
Now, don't notice this was 3 times x plus 2.
We gave it a factor of y, so it was just like this.
We have to give that a factor of y.
So you can leave your answer just like that.
This value, if it's the quantity 4x plus 1 times y, it
will be a true statement.
This has been section 5.2 part one, dealing with addition and
subtraction of rational expressions.
We'll see you next time in 5.2 part two.
Thank you.