Chapter 2 Lesson 3
Variable Rules
How to multiply a
monomial times a binomial.
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Let's say you multiply
a monomial and binomial.
You know, I think I
need a reality tv show.
What does that have to do
with monomials and binomials?
That's what reality tv does.
Put different people together
to see what happens.
What was I thinking?
Put some hot babes in a hot
tub and you have a hot tv show.
I could be a millionaire.
2(z + 5)
That should be interesting.
Back to this reality.
What do parentheses do?
2(z + 5)
Multiply. But the binomial
is too big to multiply.
It's like me against a sump wrestler.
2(z + 5)
Multiply each term 1 at a time.
2(z + 5)
You mean, like 1 at a time?
2(z + 5)
Multiply the terms
one at a time.
It goes like this. One at a time.
2(z + 5)
2z
10
What happens to the sign?
2(z + 5)
2z + 10
It stays right there.
That's 2z + 10.
- 2a(a - 5)
What happens with negatives?
- 2a(a - 5)
What always happens
with negatives?
- 2a(a - 5)
My advice. Multiply 1 at a time.
That's - 2a times a.
What does that equal?
- 2a(a - 5)
- 2a
2
It's - 2 A squared.
Now, multiply - 2a times - 5.
- 2a(a - 5)
2
- 2a + 10
There it is. Double negative.
Just like all the other negatives.
Binomials
So, what does that tell
you about binomials?
Binomials are altogether,
even when you multiply.
You like this rule, don't you?
3(x + 2)
There are a few things you
have to get, or you are lost.
This is one of them.
3(x + 2)
I'm starting to get the idea.
Oh, I know the answer.
3(x + 2) = 3x + 6
You have to multiply
3 times X + 2, because...
3(x + 2) = 3x + 6
Don't tell me.
Binomials act altogether.
Sorry if I sound like a recording.
Qs
2(5 + z)
How do you multiply 1 term
and a binomial?
2(5 + z)
10 + 2z
Multiply one at a time.
- 2(3a - b)
What rule multiplies
a binomial with negatives?
- 2(3a - b)
2 negatives is a positive.
- 6a + 2b
2
3x(2x - 1)
What's the 1st step to multiply this?
2
3x(2x - 1)
6x
3
Multiply 1st terms.
What's the 2nd step?
2
3x(2x - 1)
3
6x - 3x
Multiply the 2nd term.
Chapter 2 Lesson 3a
Variable Rules
Factor by a common factor.
Here's a word you should
remember from along time ago.
What is a factor?
We're playing football and our
coach says the rain is a factor.
Oooh, that's a good idea.
Math uses the same thing.
I just know I fumbled twice
and coach was mad at me.
How does algebra use it?
2 x 5 = 10
In math, it's the numbers
that multiply to get a number.
Now we'll go backwards.
15
What do you think
about to factor a number?
15
Find what multiplies to get it.
3 x 5 = 15
15 5 = 3
Or you can divide to get it.
Either way, 3 x 5 are the factors.
3 x 5 = 15
15 5 = 3
You know, this isn't 5th
grade math. I can do this.
(6x + 3)
So, how do you find the
factors for a binomial?
(6x + 3)
I'd find the number that both
of these terms multiply with.
(6x + 3)
Find the same factor
for both terms.
Well, that's a new rule.
They both divide by 3.
(6x + 3)
Find the same factor
for both terms.
Ok, pull it outside the binomial.
6x + 3
3(2x) + 3(1)
3(? + ?)
Ok, 3 is outside the binomial.
What happens next?
6x + 3
3(2x + 1)
It left 2x and 1 inside.
The 3 actually has a name.
3(2x + 1)
Common Factor
3 is called the common factor.
3(2x + 1)
Common Factor
What does that mean?
Same Factors
Common means Same Factors.
There is a math name for all this.
Distributive Property
What does that mean?
They distribute halloween candy,
so it changes 1 things to many.
I'll change the problem alittle.
- 3x + 9
What happens when
you factor a negative out?
- 3x + 9
I think you factor a negative.
Hey, that made sense.
- 3x + 9
- 3(x 3)
Make that a negative 3.
Yeah, but what sign is left?
- 3x + 9
- 3(x - 3)
It left a negative? How's that?
- 3x + 9
- 3(x - 3)
What does 2 negatives
multiply to get?
- 3x + 9
- 3(x - 3)
2 negatives make a positive.
- 3x + 9
- 3(x - 3)
The same thing happens here.
15 = 3 x 5
So, factoring a number and
a binomial act the same way.
15 = 3 x 5
Think about it.
How can rain be a factor?
Rain x Slippery = Fumble
Rain x Slippery equals fumble.
Factors make things happen.
Well, did you win or lose?
Neither. Some kid broke his
arm and they stopped the game.
Ouch. At least it wasn't you.
Yeah, that's what my mom said.
I just got my butt kicked.
Qs
10
How do you factor a number?
2 x 5 = 10
It's the numbers that
multiply to get a number.
6x + 3
How do you factor a binomial?
6x + 3
3(x + 2)
Find the same factor
for both terms.
Pull the 3 outside.
Leave what's left behind.
- 3x + 9
How do you factor a negative out?
- 3x + 9
- 3(x - 3)
Take a negative out.
Watch for double negatives.
2 (x + 1)
Both terms have a 2.
What is the name for the 2?
2 (x + 1)
Common Factor
Chapter 2 Lesson 3
Variable Rules
Short Lesson
Multiply times a binomial,
then factor a binomial.
Multiply a Binomial
How do you multiply a binomial?
We'll get one and find out.
3(a + 4)
Here's a monomial
times a binomial.
What multiplies times what?
3(a + 4)
Multiply the terms
one at a time.
What will that equal?
3(a + 4)
3a + 12
12 is positive, so it's added.
2
- 2b (b - 3)
What happens with negatives?
Watch the signs this time.
2
- 2b (b - 3)
3
- 2b + 6b
2
Double negative is a positive.
Use the MA Rule for the exponents.
Now we'll go backwards.
Factor
How do you factor a number?
10 = 2 x 5
Factors multiply to a number.
How does a binomial use that?
2x + 8
Factor this binomial.
What do you think about?
2x + 8
2(? + ?)
Find the same factor
for both terms.
What is left inside?
2x + 8
2(x + 4)
Pull the 2 outside.
Leave what's left behind.
The 2 has a name.
2(x + 4)
2 is the common factor.
Common means they both
have it. Here's another one.
- 3x + 9
What happens if you factor
a negative out of a binomial?
- 3x + 9
- 3(x 3)
Negative 3 is outside.
What is left inside the parentheses?
- 3x + 6
- 3(x - 2)
It left a negative 2.
- 3 times - 2 is positive 6.
Questions
2(z + 5)
How do you multiply a
monomial with a binomial?
2(z + 5)
2z + 10
Multiply the terms
one at a time.
2
10x + 35x
What's the 1st step to
factor a binomial?
2
10x + 35x
5x( ? + ? )
Find what both terms have.
What is left in the binomial?
2
10x + 35x
5x(2x + 7)
Pull the 5x outside.
Leave what's left behind.
Check it. Multiply backwards.
Chapter 2 Lesson 3
Variable Rules
Practice #1
Multiply times a binomial,
then factor a binomial.
2(z + 5)
How do you multiply a
monomial with a binomial?
2(z + 5)
2z + 10
Multiply one at a time.
2
10x + 35x
What's the 1st step to
factor a binomial?
2
10x + 35x
5x( ? + ? )
Find a fact for both terms.
What is left in the binomial?
2
10x + 35x
5x(2x + 7)
Check it. Multiply backwards.
- 3x + 9
How do you factor a negative out?
- 3x + 9
- 3(x - 3)
Take a negative out.
Watch for double negatives.
Problems
- 2a(a - 3)
Find the answer.
What rule multiplies a binomial?
- 2a(a - 3)
2
- 2a + 6a
Multiply the terms
one at a time.
- 2x(- x - 1)
Multiply with negatives.
- 2x(- x - 1)
2
2x + 2x
- 2x times - x is
a double negative.
2
2a (2a - 3)
Multiply the terms.
What does it equal?
2
2a (2a - 3)
3
4a - 6a
2
3
2
- a (2a + 1)
Multiply the terms.
What does it equal?
3
2
- a (2a + 1)
6
- 2a - a
3
2a(b + 1)
Multiply the terms.
What does it equal?
2a(b + 1)
2ab + 2a
Multiply the terms
one at a time.
Chapter 2 Lesson 3
Variable Rules
Practice #2
Multiply times a binomial,
then factor a binomial.
2(z + 5)
How do you multiply a
monomial with a binomial?
2(z + 5)
2z + 10
Multiply one at a time.
2
10x + 35x
What's the 1st step to
factor a binomial?
2
10x + 35x
5x( ? + ? )
Find a fact for both terms.
What is left in the binomial?
2
10x + 35x
5x(2x + 7)
Check it. Multiply backwards.
- 3x + 9
How do you factor a negative out?
- 3x + 9
- 3(x - 3)
Take a negative out.
Watch for double negatives.
Problems
2
2x - 12x
Factor this binomial.
What's the 1st step?
2
2x - 12x
2x(? - ?)
Take 2x out of each term.
What is left over?
2
2x - 12x
2x(x - 6)
It leaves X - 6 inside.
3a + 12
Factor this, all in 1 step.
What's the answer?
3a + 12
3(a + 4)
Take 3 out of each term.
It leaves the negative.
- 6c + 3
Factor a negative out.
What is left behind?
- 6c + 3
- 3(2c - 1)
- 3 times - 1 is
a double negative
3
4
- 3c (2c - 1)
Multiply the terms.
What does it equal?
3
4
- 3c (2c - 1)
7
- 6c + 3c
3
Double negative makes it positive.
2
2
- x (2x + 2)
Multiply the terms.
What does it equal?
2
2
- x (2x + 2)
4
- 2x - 2x
2
The d does not change.
Chapter 2 Lesson 3
Variable Rules
Performance #1
Multiply times a binomial,
then factor a binomial.
2(4a + 3)
3(2a - 5)
Multiply each binomial.
2(4a + 3)
8a + 6
3(2a - 5)
6a - 15
- 2(4x - 3)
- 4(2x - 1)
Multiply binomials.
- 2(4x - 3)
- 8x + 6
- 4(2x - 1)
- 8x + 4
2
2a (4a + 3)
Multiply with an exponent.
2
2a (4a + 3)
3
8a + 6a
2
2
2c (- 2c - 1)
Multiply with negatives.
2
2c (- 2c - 1)
3
- 4c - 2c
2
(6x + 2)
(9a - 3)
Factor the binomials.
(6x + 2)
2(3x + 1)
(9a - 3)
3(3a - 1)
(- 6x + 3)
(- 4b - 2)
Factor a negative out.
(- 6x + 3)
- 3(2x - 1)
(- 4b - 2)
- 2(2b + 1)
2
(- 2x - x)
Factor the binomial.
2
(- 2x - x)
- x(2x + 1)
2
(6c - 2c)
Factor the binomial.
2
(6c - 2c)
2c(3c - 1)
Chapter 2 Lesson 3
Variable Rules
Performance #2
Multiply times a binomial,
then factor a binomial.
2(5x + 2)
- 3(2x - 4)
Multiply each binomial.
2(5x + 2)
10x + 4
- 3(2x - 4)
- 6x + 12
- 3(- x - 3)
- 8(- x - 1)
Multiply binomials.
- 3(- x - 3)
3x + 9
- 8(- x - 1)
8x + 4
2
3a (2a + 1)
Multiply with an exponent.
2
3a (2a + 1)
3
6a + 3a
2
4c (- 2c - 2)
Multiply with negatives.
4c (- 2c - 2)
2
- 8c - 8c
(6x + 9)
(4a - 2)
Factor the binomials.
(6x + 9)
3(2x + 3)
(4a - 2)
2(2a - 1)
(- 6x - 4)
(- 4b - 4)
Factor a negative out.
(- 6x - 4)
- 2(3x + 2)
(- 4b - 4)
- 4(b + 1)
3
2
(- 2x - x )
Factor the binomial.
3
2
(- 2x - x )
2
- x (2x + 1)
2
(6c - 4c)
Factor the binomial.
2
(6c - 4c)
2c(3c - 2)