Algebra II: Simplifying Polynomials

17-22, simplifying polynomials and algebraic expressions

We're on problem 17.

And it asks us which expression
shows the complete

factorization of 12x
squared minus 147?

Let's see, first of all,
can we factor out?

147 is a strange
looking number.

Let's see if we can factor
out something.

It's definitely not divisible
by 12, because

12 goes into 144.

Is it divisible by 3?

1 plus 4 is 5, 5 plus 7 is 12,
and 12 is divisible by 3.

And I don't know if you've
seen that before.

You just add up the digits and
if that is divisible by 3,

then the number's
divisible by 3.

So 147 is the divisible by 3.

Let's see what it is.

Let's work it out.

It's 4 times 3 is
12, 27, it's 49.

So we can factor
3 out of this.

So then it becomes 3 times--
what's 12 divided by 3-- 4x

squared minus-- 147 divided
by 3, which we just

figured out, was 49.

And now this, once again, looks
just like a squared

minus b squared, where a would
be 2x and b would be 9.

So now we can factor
that more.

And I'll just switch
colors arbitrarily.

That becomes 3 times a plus b,
so that would be 2x plus 7

times a minus b.

2x minus 7.

And that is choice D.

In choice D, they wrote the 2x
minus 7 first, but it's the

same difference.

So it's choice D.

Problem 18.

Let me see if I can get
problem 18 going.

Problem 18.

Well, maybe I should
cut and paste it.

Why not?

So I copied it and then
let me paste it here.

That's the problem.

I'll rewrite it because I don't
if that's big enough for

you to see.

So it says x plus 3 over x plus
5 plus 6 over x squared

plus 3x minus 10.

So when you add fractions,
whether you're doing it with

algebraic fractions or regular
fractions, you have to find a

common denominator.

And let's see, we have to find
the least common multiple of

the denominators here, but I
have a suspicion that this x

plus 5 goes into this.

So let's see if I
can factor this.

What two numbers, when I
multiply them, equal minus 10,

and when I add them
equal plus 3?

Let's see.

Well, 5 times what
is minus 10?

It's 5 times minus 2 and 5 plus
minus 2 is 3, so that

works. x minus 2.

So this is actually our
least common multiple.

This expression right here.

So let me write it that way.

So that is equal to-- I'm just
going to rewrite this as x

plus 5 over x minus 2.

Now, x plus 3 over x plus 5, if
we were to multiply both of

those times x minus 2 to get
something in this form, this x

plus 3 over x plus 5 is the same
thing as x plus 3 times x

minus 2 over x plus 5 times
x minus 2, right?

You could just cancel this
out right here and

you'd get back to that.

And now we're adding that.

We're adding this term
to this term.

6 over this, well, this is the
same thing as this, so that's

just plus 6.

And now we just get into
simplification mode.

So x times x is x squared.

x times minus 2 is minus 2x.

3 times x is plus 3x.

3 times minus 2 is minus 6, so
that's this term right here.

Got us that.

And then we have the plus 6.

And then all of that
is over this stuff.

And I look at the choices and it
seems like they have it in

this form, so I'll just
write it in that form.

x squared plus 3x minus 10.

And let's see, the minus 6 and
the plus 6 cancels out, and

we're left with minus
2x plus 3x.

So that's x squared-- minus 2
plus 3-- plus x over x squared

plus 3x minus 10.

And that is choice A.

Choice A.

Next problem.

I'm almost out of space.

I'll draw a line here, just so
you don't get distracted.

What is the simplified form of,
and they write 3a squared

b to the third, c to the minus
2, all of that over--

interesting-- a to the minus 1
b squared c, and all of that

is to the third power.

So let's get in simplification
mode.

So this bottom part, we can
re-simplify as-- let's see,

maybe I wrote too big.

3a squared, b to the third,
c to the minus 2.

All of that, it's a to the
negative 1, b squared, c to

the third power, that's
each of these items

to the third power.

So a to the minus 1 to
the third power.

You can multiply
the exponents.

That becomes a to the minus 3.

I just took the negative
1 times the 3.

b squared to the third power.

That's b to the 2 times 3.

That's b to the sixth power.

And then finally c.

Well, that's just c to the first
of the third power, so

that's c to the third.

And now we can just say, well,
this is the same thing.

Let me switch colors.

This color's getting mundane.

This is equal to 3 times a to
the 2-- we're dividing by a to

the negative third-- so
it's 2 minus minus 3.

Let me write that, just so you
understand-- 2 minus minus 3

power-- that's where I got the
2; that's where I got the

minus 3, and I subtracted
because I'm dividing-- times b

to the 3 minus 6, times c
to the minus 2 minus 3.

Once again, if we were
multiplying these two, I would

add the exponents.

But anyway, let's
simplify this.

This equals 3a, 2 minus minus,
so that's plus, 3a to the

fifth, b to the minus
3, c to the minus 5.

And this is the same thing
as 3a to the fifth.

b to the minus 3 is the same
thing as 1 over b to the

third, so over b to the third.

And this is the same thing
as 1 over c to the fifth.

And that is choice A.

Choice A.

Next problem.

Next problem.

Oh, we already finished
that page.

All right.

Let me copy and paste
what they wrote.

Let me put it at the top of this
right there and paste it.

That's what they're asking
and I'll write it.

20x to the minus fourth
over 27y squared

divided by this fraction.

So first of all, when you divide
by a fraction, that's

the same thing as multiplying
by the inverse, right?

So let's do that.

I want to get rid of this
pesky-looking division sign.

So let's rewrite this as 20x to
the minus fourth over 27y

squared times-- instead of
dividing by this, let's

multiply by the inverse.

So what's the inverse?

15y to the minus 5 over
8x to the minus 3.

I just flipped it.

All right?

So let's see what
we can do here.

It seems like a lot of these
numbers have common factors.

Let's see, if we divide 15
by 3 and 27 by 3, so

this becomes 5.

27 divided by 3 becomes 9.

And just so you can think,
you can view this as one

continuous denominator or
one continuous fraction.

20x to the minus 4 times-- well,
now it's 5y to the minus

fifth divided by 9y squared.

Because when you multiply
fractions, you just multiply

the numerator times the
numerator divided by the

denominator times
the denominator.

Anyway, let's just
keep simplifying.

If you divide this
by 4, you get 5.

If you divide this
by 4, you get 2.

Let's see, I don't want to do
too many steps all at once.

You get 5x to the minus fourth
times 5y to the minus fifth,

all of that over 9y squared
times 2x to the minus 3.

So let's see.

Let's get all the numbers out.

So that is equal to 5 times 5
is 25 over 9 times 2 is 18

times x-- let's do the x-- x to
the minus 4-- minus because

we're dividing-- minus 3.

Minus minus 3 times y to the
minus 5 minus 2, because we're

dividing by that one.

And that is equal to 25/18.

Let's see, a minus minus,
so that becomes a plus.

Minus 4 plus 3, x to the
negative 1, and then minus 5

minus 2, y to the minus 7.

This is the same thing
as 1 over x to the 1.

This is the same thing
is 1 over y to the 7.

So this is equal to
25 over 18x, 1

over x, y to the seventh.

y to the seventh in the
denominator is the same thing

as y to the minus seventh
in the numerator.

Anyway, that is choice--
let's see, 25 over 18--

that is choice D.

Choice D.

And I'm out of time.

See you in the next video.