CA Algebra I: Rational Expressions

66-70, more simplifying of rational expressions

We're on problem 66.

And it says what is x squared
minus 4x plus 4, divided by x

squared minus 3x plus 2, reduced
to lowest terms?

So they probably want us
to factor each of these

quadratics and see if any of
these terms cancel out.

So let's try to do that.

So the numerator, this seems
pretty easy to factor.

What two numbers when I
multiply them equal 4?

And when I add them
equal minus 4?

Well it's minus 2, right?

Minus 2 and minus
2 is minus 4.

Minus 2 squared is plus 4.

So this is x minus 2
times x minus 2.

And you could test it if
you don't believe it.

Multiply that out.

Divided by, let's see,
what two numbers?

This looks factorable.

They both have to be the same
sign because when you multiply

them you get a positive.

And they're both going to be
negative, because when you add

them you get a negative 3.

So let's see, minus
2 and minus 1.

Minus 2 times minus
1 is positive 2.

Minus 2 plus minus
1 is minus 3.

So x minus 2, times x minus 1.

And if we assume that x is never
equal to 2, because that

would make this expression
undefined, we cancel that out.

You'll learn later that would
cause a hole in the graph,

because the function
is undefined there.

And you're left with minus
2 over minus 1.

And that is choice A.

Problem 67.

This is good practice.

They give a bunch of it.

They say what is-- I'll just
write it-- 12a cubed minus 20a

squared over 16a squared
plus 8a.

Reduce to lowest terms. So let's
just try to factor out

things on the top and the bottom
and see what happens.

So in the top, in the
numerator-- let me switch

colors-- both terms are
divisible by 4 and a squared.

So let's fact out
a 4a squared.

So we get 4a squared.

12 divided by 4 is a 3.

And a cubed divided by
a squared is an a.

So 12a cubed divided by
4a squared is 3a.

Minus 20-- I could say
plus minus 20--

but you get the idea.

20 divided by 4 is 5.

And a squared divided by
a squared is just a.

And if you don't believe
this, multiply it out.

4a squared times 3a
is 12a cubed.

And 4a squared times minus
5 is minus 20a squared.

So it works out.

You do the denominator.

Let's see, both of these are
divisible by 8a, so let's

factor that out.

16 divided by 8 is 2.

a squared divided by a is a.

So 16a squared divided
by 8s is 2a.

And if you go the other
way, 8a times 2a

squared is 16a squared.

So it all works out.

Plus 1.

8a times 1 is 8a.

So let's see what we
could do here.

This is becomes a 1.

This becomes a 2.

And a squared divided by a, this
becomes a 1 and this just

becomes just an a.

And we're left with a times
3a minus 5, over 2

times 2a plus 1.

And let's see.

That is choice D.

I thought maybe they'd want us
to re-multiply this out again.

But that is choice D.

Problem 68.

Oh this is a good one.

I'll just write it.

They want us to multiply
something.

So they say 7z squared plus
7z-- all of that--

over 4z plus 8.

Times z squared minus 4-- all of
that-- over z to the third

plus 2z squared plus z equals.

So you must be like oh my god,
I have to multiply all of

these things and I have
to divide them.

But the best thing, I'm
guessing, is to just factor

these out and all sorts of
things will start canceling

out with each other.

And it will turn into a
pretty simple problem.

Let's see, both of these terms
are divisible by 7z.

So le'ts factor that out.

So that top part becomes, 7z
squared divided by 7z, you

just have a z left.

If multiply these, you
get 7z squared.

Plus 1.

If you multiply this out, you
get 7z squared plus 7z.

When you multiply fractions,
it's just the numerator times

the numerator, over the
denominator times the

denominator.

So this is times
the numerator.

Z squared minus 4, that's a
squared minus b squared.

So that's z plus 2, a plus b,
times z minus 2, a minus b.

That's just the pattern when I
say all those a's and b's.

So that's z plus 2 times z minus
2, hopefully you can

recognize that at this point.

And then all of that over--
let's see, we can definitely

factor out a 4 here, so that's
4 times z plus 2.

8 divided by 4 is 2 times-- so
we can definitely factor out a

z here, so we get z times z
squared plus 2z plus 1.

I think we're almost done.

Now we have to factor this.

Let me just rewrite
everything.

So this is equal to 7z times
z plus 1, times z plus 2,

times z minus 6.

All of that over 4 times
z plus 2, times z.

And what's this?

This is z plus 1 squared.

Z plus 1 times z plus
1, 1 times 1 is 1,

and 1 plus 1 is 2.

So times z plus 1,
times z plus 1.

And now is fun part.

This is a 1 here, that's
parentheses.

Now we can start
canceling out.

And we assume that the
denominator would never equal

0 and all that.

Let's see, this z plus
2 cancels out

with this z plus 2.

This z plus 1 cancels out with
one of these z plus 1's.

I'll do the one that's
written messier.

And let's see, this z cancels
out with this z.

And what are we left with?

Everything simplified to 7
times z minus 6 over 4

times z plus 1.

I wrote a z minus b here.

It's z plus 2 times z minus 2.

All that pattern matching,
I made a mistake.

Z squared minus 4 is z plus
2 times z minus 2.

Not z minus b, and I though
that was a 6.

So this is z minus 2.

So this is z minus 2.

And so that is choice A.

Sorry about that error.

Brain malfunctions
all the time.

All right, now they want
us to do it again.

They want us to find the product
of x plus 5, over 3x

plus 2, times 2x minus
3, over x minus 5.

Frankly, there's not a lot of
simplification we can do, we

just have to multiply it out.

So this is going to be equal to
x plus 5 times 2x minus 3.

All of that over 3x--

I'm just multiplying the
numerator and then multiplying

the denominators-- 3x plus
2 times x minus 5.

And now we just multiply
both binomials, x

times 2x, 2x squared.

X times minus 3, minus 3x.

5 times 2x, plus 10x.

5 times minus 3, minus 15.

Fair enough.

Now you do the denominator.

3x times x is 3x squared.

3x times minus 5, minus 15x.

2 times x, plus 2x.

2 times minus 5, minus 10.

And now let's see if
we can simplify.

We have the numerator is
equal to 2x squared

minus 3x plus 10x.

So that's plus 7x minus 15.

All of that over 3x squared.

And minus 15x plus 2x.

That's minus 13x minus 10.

And that is choice D.

Next problem.

Problem 70.

Boy, they they want us
to keep this up.

This is good practice.

So they write, x squared plus
8x plus 16, over x plus 3,

divided by 2x plus 8, over
x squared minus 9.

So the first thing you do, when
you divide by a fraction,

it's the same thing is
multiplying by its inverse.

So this is equal to x squared
plus 8x plus 16, over x plus 3

times the inverse of this,
x squared minus 9,

over 2x plus 8.

Fair enough.

Now let's see if we can simplify
these a little bit.

I'll do that in yellow.

So this is, 4 plus 4 is
8, 4 times 4 is 16.

So this we can rewrite as
x plus 4 times x plus 4.

x squared minus 9, that's a
squared minus b squared.

So this we can rewrite as x
plus 3 times x minus 3.

It's going with the pattern.

We can factor out a 2 here, so
we can rewrite this as 2

times x plus 4.

We have an x plus 3 there.

And of course, when we multiply
fractions, we're just

multiplying all the numerators
over all the denominators.

So it's almost like you
make this one line.

So the numerator is x plus 4
times x plus 4 times x plus 3

times x minus 3.

All of that over x plus 3
times 2 times x plus 4.

So now let's do some
cancellation.

This is the fun part.

So we have an x plus 4 and and
x plus 4, cancel them out.

We have an x plus 3 and an x
plus 3, cancel them out.

And what are we left with?

We are left with an x plus
4 times an x minus 3.

All of that over 2.

And that is choice C.

And I will see you in
the next video.