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>> Professor Gracey: Hi there,
this is Professor Shannon
Gracey from MiraCosta College
and we are now covering
section 7.3
in Robert Blitzer's
Introductory
and Intermediate Algebra
for College Students text
and this is our very last
section for our summer course.
So here we go.
We will be adding
and subtracting rational
expressions
with the same denominator
and then we'll also look
at adding and subtracting
rational expressions
with opposite
denominators today.
So let's warm up.
The first warm-up has
us simplifying.
And pause the movie
and see how you do.
Okay, let's see how you did.
If you take a look here we
almost have the same factors.
So if you recall you can
either factor out a negative
or if you don't
like the factoring
out the negative business what
you can do is you can multiply
both numerator and denominator
by a negative one
over a negative one.
And the trick is you don't
multiple both negative
ones throughout.
What you do is you pick one
of them.
So I'll go ahead
and multiply the negative
throughout the numerator
and notice we'll end
up getting negative B squared
plus A squared
over A squared minus B squared
and then we still have the
negative 1 hanging out.
Now commuting the terms
in the numerator do you see we
get A squared minus B squared?
And it's true
to that one will factor
but it's already the same
as this guy.
So you know what?
Why don't we make your lives
easier and go ahead
and divide out here?
And at the end
of the day we will get 1
over negative 1 which ends
up giving us a result
of negative 1.
So that rational expression
simplifies to be negative 1.
How did you do?
Awesome. All right,
let's go on to the next guy.
Okay so the numerator will
factor as X minus one,
the quantity squared.
We have a perfect square
trinomial or if you factor
using the X you probably got X
minus 1 times X minus 1.
Now let's look here
at the denominator.
This one I'm going
to do this one a
little differently.
We're going to factor
out a negative
so that way you can decide
which way you like better.
So if I factor
out a negative 1 do you see
I'll get negative 1 plus X?
So this is how the numerator
factored and the denominator
factors like this.
Well, this is equivalent
to X minus 1,
the quantity squared
over negative 1 times X
minus 1.
Now notice that's
like X minus 1 to the 1
so what will happen is,
let's do it
like this this time.
We will have a negative
from that negative 1
and we will end
up with X minus 1
to the 2 minus 1 using the
properties for exponents.
We're dividing the same base
so we subtract the exponents.
There's a power of 1
on this guy so we will end
up with a result of X minus 1
to the 1 and that's the
opposite of that,
so you can either put your
result as the opposite
of X minus 1
or you could have put negative
X plus 1.
That's up to you.
All right, so onward.
If P/R and Q/R are rational
expressions then P/R plus Q/R,
notice they have the same
denominator,
so we can add the numerators
and put them
over that common denominator.
So to add rational expressions
with the same denominator add
numerators and place the sum
over the common denominator.
If possibly,
simplify the result.
So we're attacking this a
little bit differently.
When you're adding rational
expressions you wait
on that factoring step.
So if you recall,
when simplifying rational
expressions,
multiplying rational
expressions,
and dividing rational
expressions we factored very
early in the problem.
Now here we first need
to find the sum or difference,
depending on what you have,
and then you compute that sum
and then you simplify
as your last step.
So here we go subtracting,
so if we have, again,
P/R and Q/R
and they're rational
expressions then P/R minus Q/R
give us P minus Q all over R.
So just subtract rational
expressions
with the same denominator.
Subtract numerators
and place the difference
over the common denominator
and then at the end simplify
the result.
Okay, so let's take a look
at our first example.
Add or subtract as indicated.
So first up you check.
You make sure you have the
same denominator.
So here we go.
Part A, each denominator
is 15.
So first off put your common
denominator 15.
We do not add denominators.
We put that common denominator
just that once then we're
adding the numerator,
the two numerators
so 1X plus 4X is 5X/15
and now notice we
can simplify.
Five goes into itself once
and into 15 three times.
So at the end
of the day we get 1,
times X is X over 3.
And we're done.
All right,
so you guys go ahead
and pause the movie
and you try B, C,
and D and see how you do.
Okay, let's see how it went.
So here we go.
In Part B we have a common
denominator of 9.
We are adding X plus 4 plus 2X
minus 25.
Now notice I don't need those
parentheses
so I'll take them off
and that's all over 9
and then combining
like terms I'll get 3Xs minus
21 all over 9.
I'll just go down here.
I can factor out a 3
from the numerator
and I'll be left
with X minus 7
and that's all over 9.
So this is how I'm
on the simplifying step
and that's when you factor
and notice 3 goes
into itself once
and into 9 three times,
so my end result will be X
minus 7 all over 3.
How did you do?
I'll bet did you great.
Okay, Part C. Here we go.
Checking my denominator I've
got the same denominator.
I'm going to go ahead
and write that common
denominator
down then I am taking X minus
1, so that is merely X minus 1
over X minus 1,
so X minus one goes
into itself once.
So believe it
or not this simplifies
to be 1/1 which is 1.
All right,
Part D we have a common
denominator of 3X plus 4
so we're going to put
that down right away
and then we're going
to be adding 3Xs plus 2
and then plus 3X plus 6.
Well we don't need those
parentheses so I'll rewrite it
like this and then I end
up with 6X plus 8
over 3X plus 4.
I can factor out a 2
from the numerator
and I'm left
with 3X plus 4 that's
over 3X plus 4 so this here,
this is the part
where we factor
so we can simplify.
3X plus 4 goes
into itself one time,
so at the end
of the day this simplified
to be 2/1 which is 2.
Now why don't you go ahead
and pause the movie
and give E, F, and G a try?
All right let's see how
you did.
So here we go.
I am looking at the fact
that we have a common
denominator of 2X
to the fourth.
Now we're subtracting
so we've got X cubed minus 3
minus 7X cubed minus 3.
So X cubed minus 3,
I have to distribute the minus
throughout the
second parentheses.
Oops, sorry.
I was about to make
up my own math there.
That will be a plus 3
and then we have our common
denominator of 2X
to the fourth
so then we'll get let's see,
negative 6X cubed
and do you see it will be
negative 3 plus 3 is zero
and then we have 2X
to the fourth,
so then we will get negative 6
divided by 2 is negative 3
and then we'll get X
to the 3 minus 4
which is negative 3X
to the negative 1
which is negative 3/X.
So just to clarify you know,
I got to negative 3 from that
and then using the properties
for exponents is how I did
this guy here.
So next up Part F we have the
same denominator.
This denominator is 4X squared
minus 11Xs minus 3
and then we're going
to have X squared plus 9Xs
plus 3X minus 5X squared this
will be X squared plus 9X plus
3X minus 5X squared
over 4X squared minus 11X
minus 3.
We will get negative 4X
squared plus 12X over--
now I'm going to take a look
at factoring the denominator
so let's see.
We have an overall product
of negative 12 so
and then a middle term
of negative 11,
that co-efficient,
so I believe
if we did negative 12
and positive 1 those numbers
would work.
So 4X squared minus 12Xs plus
1X and then minus 3
so here I've got this is
equivalent to this so far
and then you do your red group
and your blue group
for your factoring
and then we will have,
if we factor
out from the numerator I can
factor out a negative 4X
and I'm left with 3X,
I am left with X minus 3.
So this is the way,
I'm on the simplifying portion
of it.
This is the way
that the numerator can factor
and then I can factor out a 4X
and I'm left X minus 3
and plus 1 times X minus 3.
So this is the continuation
of the factoring
of the denominator.
Carrying on the factored
numerator I can factor
out an X minus 3 and I'm left
with 4X plus 1.
So this is the completely
factored denominator,
and then take a look X minus 3
divides out our numerator
and denominator
and then our end result is
negative 4X over 4X plus 1
and we are done.
Okay, let's look
at this next guy.
So we have a common
denominator
of 3Y squared plus 10Y
minus 8.
And then we have 3Y squared
minus 2 minus quantity Y plus
10 minus quantity Y squared
minus 6Ys,
so we will get 3Y squared
minus 2 minus Y minus 10.
So I'm distributing that minus
through minus Y squared plus
6Y all over 3Y squared plus
10Y minus 8.
So this will give us a result
of 2Y squared
and so we have a minus Y
and a positive 6Y so plus 5Y
and then minus 12 all
over 3Y squared plus 10Y
minus 8.
Now we need to see
if these guys factor,
so if we do our Xs,
I'll go ahead and put them
in down here,
so for the numerator overall
product of negative 24
and then a middle term
of positive 5
so we need a difference
that results in positive 5.
So I believe the two numbers
that fit the bill are negative
3 and positive 8.
So we will have 2Y squared
minus 3Y plus 8Y minus 12
over, just doing the factoring
for the denominator,
overall product
of negative 24,
the 3 times the negative 8
and then let's see,
middle term of positive 10.
That's the results
of the difference.
I believe the two numbers
that fit the bill are negative
2 and positive 12
so we will have 3Y squared
minus 2Y plus 12Y minus 8.
So the numerator
on the factoring part
because I'm simplifying
and this is the start
of the denominator,
so here we go.
And again,
this is all the whole
factoring by grouping business
so we're using all our
wonderful skills,
so in the numerator
if I factor out a Y I'm left
with 2Y minus 3 and then
if I factor
out a positive 4 I'm left
with 2Y minus 3 all over,
let's see in the denominator I
can factor out a Y. I'm left
with 3Y minus 2
and if I factor
out a positive 4 I'm left
with 3Y minus 2.
So the yellow is on its way
to being factored
as is the blue.
Okay, next
up I have a common factor
of 2Y minus 3 in the numerator
and I'm left with Y plus 4
so now the numerator is
completely factored
and in the denominator I can
factor out 3Y minus 2
and I'm left with Y plus 4
so the denominator is now
fully factored
and I can divide
out the Y plus 4 factor
and at the end
of the day I am left
with 2Y minus 3
over 3Y minus 2.
And we're all set.
All right,
so our very last task is
to be able to add
and subtract rational
expressions
that have
opposite denominators.
So when one denominator is the
opposite or additive inverse
of the other first multiply
either rational expression
by negative 1 over, oh,
let me write
that a little better,
by negative 1 over negative 1
to obtain a
common denominator.
Remember the denominators need
to be the same in order to add
or subtract.
So here we go.
So this first one,
notice that the denominators
are almost the same they're
just opposite.
So what I'm going
to do is go ahead
and multiply the second one
by a negative 1
over negative 1
so then what will happen is we
will get 6X plus 7
over X minus 6
and then plus negative 3X
over, now notice
that well I'll write it
out so we'll have negative 6
plus X when I distribute
that negative 1 throughout the
second denominator.
This give us 6X plus 7
over X minus 6 plus negative
3Xs over X minus 6
if I commute those terms.
Now I have a common
denominator and it's just
like the other stuff we
were doing.
So now putting the common
denominator
down as my denominator I will
have 6X plus 7 plus negative
3X which gives me 6X minus 3X
is positive 3X plus 7
over X minus 6.
Both of these are,
the numerator
and the denominator are
prime polynomials.
So we're all set, we're done.
Okay? Okay, so go ahead
and pause the movie
and you guys try B, C,
and D and then we'll be all a
done with section 7.3
and with our MiraCosta
beginning algebra course.
Okay, let's see how you did
on Part B. Again,
I'm going to go ahead
and multiply the second
denominator by negative 1
so then this will give us X
squared over X minus 3 plus
negative 9
over negative 3 plus X. We
will have X squared
over X minus 3 plus negative 9
over X minus 3.
Now we have the same
denominator
so do you see we will have X
squared minus 9
over X minus 3?
But I can factor
that difference of squares
in the numerator
to be X plus 3 times X minus 3
and that's over X minus 3.
So this guy factors to this
and then we can divide
out that common factor
of X minus 3 and we're left
with X plus 3 over 1
which is just X plus 3.
How did you do?
Awesome. I bet you did
really great.
Okay so Part C up here, again,
let's do that second fraction
and I'll show you a little
trick here.
Now this will give us,
I think I have room
on the side,
so this will give us 4 minus X
over X minus 9.
Now here's the trick.
Do you see here we've got
minus a minus?
So instead of distributing
that negative 1
through the 3X minus 8
and then distributing it again
when you subtract I'm going
to turn that minus a minus
into a plus
so this is the result
of the minus a minus.
Then we'll have 3X minus 8
over negative 9 plus X.
So this is going
to give us 4 minus X
over X minus 9 plus 3X minus 8
over X minus 9
which will give us 4 minus X
plus 3Xs minus 8 all
over that common denominator
of X minus 9.
So then negative X plus 3X is
2Xs and 4 minus 8 is negative
4 and then we'll have
that over X minus 9
and then factoring the
numerator we get 3 times
quantity X minus 2
over X minus 9.
There's no common factors
so this is our final result.
Okay and Part D again,
multiplying by negative 1
over negative 1
on that second fraction we
will have 2X plus 3
over X squared minus X minus
30 and then plus negative X
plus 2 over negative 30 minus
X plus X squared
when I distribute
that minus through.
This will give us 2X plus 3
over X squared minus X minus
30 plus negative X plus 2
over, I'll commute those terms
and get X squared minus X
minus 30.
I now have the same
denominator so I can add
and I will have this
result here.
So then once I combine my
terms to X minus X is 1X
and 3 plus 2 is 5
and then I'm going to factor
that denominator
and do you see
that we can factor
that denominator
as leave it a factor
as X plus 5 times X minus 6?
So here's the factored form
of the denominator
which will give us the result
of, we can divide
out the X minus 5s
and then we will get 1.
Don't forget the 1
in the numerator
over X minus 6
and we are all done.
So have a wonderful rest
of your day
and a wonderful rest
of your summer.
Bye.