The Distributive Property (M2 2.1 Notes)

Summary

This educational video script focuses on polynomials, emphasizing the distributive property for multiplying and factoring. It covers combining like terms, expanding on the distributive property, and applying it to binomials. The script also introduces the zero product property for solving equations. Each concept is explained with examples to ensure a thorough understanding before moving on to the next topic.

Takeaways

• 📚 The lesson focuses on polynomials, specifically on multiplying and factoring them, with an emphasis on the distributive property.
• 🔢 Adding and subtracting like terms is covered, where coefficients are combined without changing exponents.
• 🧩 Parentheses are discussed in the context of polynomials, highlighting that a plus sign before parentheses does not affect the terms inside, but a minus sign does.
• 🔄 The distributive property is explained as a method to interact between multiplication and addition, and it is also used in reverse for factoring.
• 🔢 When multiplying terms, exponents change, which is different from addition or subtraction where they remain the same.
• 🔍 Factoring is demonstrated by finding the greatest common factor and dividing each term by it, which can be checked by multiplying the factors.
• 📉 The zero product property is introduced as a tool for solving equations where a product of two factors equals zero, implying at least one factor must be zero.
• 🚫 The importance of understanding the underlying principles before using shortcuts is stressed to avoid common mistakes.
• 📘 The process of multiplying binomials is detailed, emphasizing the step-by-step application of the distributive property rather than relying on shortcuts.
• ✅ The lesson concludes with an example of using the zero product property to solve equations by factoring and setting each factor equal to zero.

Q & A

What is the main focus of Unit 2 in the transcript?

Unit 2 primarily focuses on polynomials, specifically on multiplying and factoring polynomials, with an emphasis on the distributive property.

How does the distributive property apply to polynomials?

The distributive property allows for the interaction between multiplication and addition in polynomials. It states that for any numbers a, b, and c, the expression a(b + c) is equivalent to ab + ac.

What is the rule for combining like terms in polynomials?

When combining like terms in polynomials, you add or subtract the coefficients while keeping the variable part unchanged. For example, x^2 + x^2 results in 2x^2.

Why are parentheses important when there's a negative sign in front of them?

Parentheses are important when there's a negative sign in front of them because the negative sign must be distributed to every term inside the parentheses, which affects the signs of the terms.

What is the difference between multiplying and adding or subtracting in terms of exponents?

When adding or subtracting, the exponents of like terms are not changed; only the coefficients are combined. However, when multiplying, the exponents are added together, as seen in expressions like x^2 * x^2 = x^(2+2) = x^4.

How does the zero product property help in solving equations?

The zero product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to solve equations by factoring and setting each factor equal to zero.

What is a common mistake made when distributing a negative sign across terms in parentheses?

A common mistake is to only distribute the negative sign to the first term inside the parentheses and not to all terms, which leads to an incorrect solution.

Why is it important to understand the distributive property before using shortcuts for multiplying binomials?

Understanding the distributive property before using shortcuts for multiplying binomials is important because it provides a solid foundation for why the shortcuts work, ensuring that the process is correctly applied and understood.

How does the process of factoring relate to the distributive property?

Factoring is essentially the reverse of using the distributive property. It involves finding the greatest common factor and expressing the polynomial as a product of its factors, which can then be expanded using the distributive property.

What is the significance of the zero product property in solving polynomial equations?

The zero product property is significant in solving polynomial equations because it allows for the identification of possible solutions by setting each factor in a factored equation to zero and solving for the variable.

Outlines

00:00

📐 Polynomials and Distributive Property

The paragraph introduces the topic of polynomials, focusing on multiplying and factoring them. It emphasizes the distributive property as the core concept, which is used for combining like terms and solving equations. The explanation covers how to add or subtract like terms without changing exponents, and the importance of distributing negative signs correctly. The paragraph also discusses the difference between adding/subtracting and multiplying/dividing in terms of how they affect exponents.

05:01

🔍 Factoring and the Distributive Property

This section delves deeper into factoring, which is presented as the reverse of the distributive property. It explains how to factor out common terms from polynomials and how to verify factoring by multiplying the factors. The paragraph highlights the importance of identifying the greatest common factor and demonstrates the process with examples. It also touches on the concept of binomials and how to apply the distributive property when multiplying them.

10:02

📘 Multiplying Binomials and Common Mistakes

The paragraph discusses the process of multiplying binomials using the distributive property, cautioning against taking shortcuts without understanding the underlying principles. It provides a step-by-step approach to multiplying binomials, emphasizing the need to distribute each term correctly. The paragraph also points out common mistakes, such as incorrectly squaring binomials, and explains the correct method to avoid these errors.

15:15

🔄 Applying the Zero Product Property

This section introduces the zero product property, which states that if the product of two factors equals zero, then at least one of the factors must be zero. The paragraph explains how this property can be used to solve equations by setting up a product equal to zero and then factoring. It provides examples of how to apply the zero product property and how it relates to the distributive property in factoring.

20:19

🧩 Homework and Further Exploration

The final paragraph briefly mentions the homework assignments that correspond to the topics covered in the script. It suggests that the homework will involve applying the concepts learned, such as the distributive property and the zero product property, to solve problems. The paragraph also hints at further exploration of these mathematical concepts in subsequent lessons.

Keywords

💡Distributive Property

The distributive property is a fundamental algebraic principle that allows for the simplification of expressions involving both multiplication and addition or subtraction. In the video, it is used to explain how to multiply a monomial by a polynomial, such as multiplying 'x' by 'x + 3' to get 'x^2 + 3x'. The property is also applied in reverse for factoring, where a polynomial is broken down into simpler expressions. For example, the script mentions factoring out the greatest common factor from a set of terms, like taking 'x' out of 'x^2' and '5x' to get 'x(x + 5)'.

💡Combining Like Terms

Combining like terms is a process in algebra where terms that have the same variables raised to the same power are added or subtracted. In the video, this concept is demonstrated when adding 'x^2' and 'x^2' to get '2x^2', or when adding '-4x' and '5x' to get 'x'. The script emphasizes that when combining like terms, the exponents remain unchanged, and only the coefficients are added or subtracted.

💡Factoring

Factoring is the process of expressing a polynomial as the product of its factors. The video script uses factoring to break down complex expressions into simpler ones, which can be easier to work with. For instance, the script discusses factoring out 'x' from 'x^2' and '5x', resulting in 'x(x + 5)'. Factoring is also related to the distributive property, as it is essentially the reverse operation, where instead of expanding an expression, we look for common factors to simplify it.

💡Exponents

Exponents, also known as powers, indicate how many times a base number is multiplied by itself. In the video, exponents are used to describe the process of multiplying terms with variables, such as 'x' times 'x' equals 'x^2'. The script explains that when multiplying terms, the exponents change according to the rules of exponents, which is a key difference from adding or subtracting like terms, where the exponents remain the same.

💡Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is used in the video to solve equations by setting up a product of two expressions equal to zero and then factoring to find the possible values of the variables. The script gives an example of setting 'x(x + 5) = 0' and then solving for 'x' by considering each factor separately.

💡Parentheses

Parentheses are used in algebraic expressions to group terms and indicate the order of operations. The video script explains that parentheses can be dropped when there is a plus sign in front of them, as the terms inside are added regardless. However, when there is a minus sign, the negative sign must be distributed to each term inside the parentheses, as shown in the script when discussing the difference between '2x^2 - 4x + 6' and '-x^2 + 5x - 3'.

💡Greatest Common Factor

The greatest common factor (GCF) is the largest factor that two or more numbers or expressions share. In the video, the GCF is used when factoring to determine the most significant factor that can be extracted from each term in a polynomial. For example, the script mentions factoring out '9x^2' from '9x^2' and '36x^2', which simplifies to 'x' after dividing each term by the GCF.

💡Binomials

A binomial is an algebraic expression with two terms, such as 'x + 3'. The video script discusses the multiplication of binomials, emphasizing the importance of understanding the underlying principles before using shortcuts. The distributive property is applied to multiply each term in one binomial by each term in another, as illustrated when multiplying 'x' by '(x + 3)' to get 'x^2 + 3x'.

💡Solving Equations

Solving equations involves finding the values of the variables that make the equation true. The video script touches on using the zero product property to solve equations by setting up expressions that equal zero and then factoring to find the solutions. This is demonstrated when the script shows how to solve 'x^2 + 5x = 0' by factoring out 'x' to get 'x(x + 5) = 0', leading to the solutions 'x = 0' or 'x = -5'.

💡Like Terms

Like terms are terms in a polynomial that have the same variables raised to the same powers. The video script explains that when combining like terms, you add or subtract the coefficients while keeping the variables and their exponents unchanged. This is shown when adding '2x^2' and 'x^2' to get '3x^2', or when adding '-4x' and '5x' to get 'x'.

Highlights

Introduction to polynomials and their operations, focusing on multiplication, factoring, and the distributive property.

Explanation of combining like terms by adding or subtracting coefficients without changing exponents.

Clarification on the irrelevance of parentheses when there is a plus sign between them, but the importance when there is a minus sign.

Instruction on correctly distributing a negative sign across terms within parentheses.

Demonstration of the distributive property in action, showing how to multiply a term by each term within parentheses.

Emphasis on the change in exponents during multiplication, as opposed to addition or subtraction.

Tutorial on factoring by identifying the greatest common factor and dividing each term by it.

Guide on checking factoring by multiplying the factored form to ensure it equals the original expression.

Discussion on factoring multiple terms by finding the greatest common factor among them.

Explanation of the zero product property and its application in solving equations.

Illustration of solving equations using the zero product property by factoring and setting each factor to zero.

Advice on not using shortcuts without understanding the underlying concepts, especially in the context of polynomial operations.

Instruction on multiplying binomials by distributing each term in the binomial to every term in the other expression.

Clarification on the common mistake of incorrectly squaring a binomial by not properly applying the distributive property.

Emphasis on the importance of correctly applying the distributive property when squaring a binomial to avoid mistakes.

Transcripts

so unit two is gonna have about

polynomials

it's about multiplying polynomials and

dividing not dividing sorry

factoring multiplying and factoring

primarily all around something called

the distributive property and there

there are some smaller components like

adding which we're going to do really

quickly and solving which is a little

bit bigger component that we're gonna

look at and those will follow us through

each section here so we're gonna start

most basic

right now and really look at the

distributive property and what it is and

then we're going to expand upon it in

future lessons so first of all combining

like terms combining where meaning

adding or subtracting like terms and

just as we did with square roots we do

here so x squared plus x squared you

have some background with what would the

answer be 2x squared so guys when you

add you never change exponents when you

add you're not adding really the x

squared and the x squared you're adding

the coefficients in front the 1 plus the

1 which gives us 2x so it's like I have

1x squared and I have another x squared

so together I have two experts okay so I

number two same idea we have parentheses

here if you have a plus sign between

parentheses this is the same problem as

if you dropped all the parentheses so

this is I could just do this 2x squared

minus 4 X 6 plus x squared 5x minus so

those parentheses are unnecessary it's

okay to put them there but they don't

have to be there now on three it's

different those parentheses do matter

really the second set does because

there's a negative sign in front of it

and that negative sign is going to have

to be distributed to everything in there

so but the plus sign doesn't matter so

much and here we look for like terms so

I got in 2x squared and an x squared if

I add those together I get 3x squared if

I have a negative 4x and a positive 5x

together those add up to 1 X and I don't

usually don't write the 1 and then if I

have a positive 6 and a negative 3 add

those together and you get positive 3

and that's as much as we can do we

cannot add the X the 3x but that's

because

- powers we have x squared here we have

expert on number three similar thing

except we have to distribute this

negative to all parts inside the second

parenthesis before we combine like terms

and that's important to pause and just

realize because a common mistake is to

do this right - x squared minus 4x plus

6 - x squared plus 5x minus 3 where this

is wrong where you're only putting the

negative in front of the x squared

that's not even distribute to the other

two terms and you'll end up with the

wrong answer so instead it should be

then fix that last part should be minus

5x and positive three switch the signs

and then you're adding like terms s

before so 2x squared and a negative x

squared gives us x squared negative 4x

and negative 5x gives us negative 9 x +

6 + 3 gives us alright so we'll use that

combining like terms as we're working

through these problems now the

distributive property again you probably

use this we just did it with the

negative sign we distributed anyway and

so the distributive property here is a

way to interact between multiplication

and addition and it's just a rule it

says hey if you have multiplication with

some addition here you can distribute

multiply a times B and a times C and I

typically put it alphabetically so a B

plus a C and that would much as I can do

I can't go any further than that that's

multiplying using the distributive

property on the on5 I multiply two x

times x squared what do I get 2 X to the

third 2x times 4 X you multiply the 2

and the 4 and get 8 you multiply x times

X and get x squared + 2 x times negative

2 negative 4x so when you multiply

that's when exponents change when you

add or subtract not so much there's a

Catedral but when you multiply or divide

then exponents start to change

you can do this same property in reverse

order it's still the distributive

property but we know the opposite of

multiplying not division but factoring

and when you're factoring like something

like number 6 you're looking at x

squared and 5x and you're asking what

can I divide both of those by what do

they both have in common and X so I'm

going to factor out an X I put that in

front and I divide each piece by X so x

squared divided by X would be X minus 5x

divided by X would be Phi and you can

always check let's see if you did it

right by multiplying it out x times X is

x squared x times negative 5 negative 5

X you know you did it right by going

backwards

number 9 what can we factor out an x

squared with a 3x squared a 9x squared a

9x so you have options but when we

factor we factor out the greatest common

factor what is the most we want to be

greedy here what is the most we can

divide both those by soap so first you

think about number 9 36 I could divide

them both by 9 and I can't do any better

than that

and then how many X's can I pull out to

from both of those so 9x squared would

be the greatest common factor and if I

divide the first part by 9x squared

would I get X because you're trying to

think 9x times what equals 9x box plus

if I divide 36 x squared by 9x squared

get 4 and again you can check this by

multiplying it out and seeing if you end

up with that answer you can do the

with three turns instead of two terms so

take a look at these three terms decide

what the greatest common factor is

factor it out and see if you can get the

right answer

you

okay what is the greatest common factor

what can you divide 28 35 and 14 by

seven goes into all of those can we do

better okay no not better with numbers

but okay now we get how many how many

X's can we divide out of all of them

yeah you're limited to the lowest one

here right you can't go above the list

how many Y's can we pull out of all of

them okay so 7 X Y squared so if I

divide each of these by 7 X Y squared I

were to do that division I'd end up with

three terms and the first one would be 4

X to the third Y the second one would be

5 y -2 X and then we've used a

distributive property in the reverse

order which we call factoring but the

distributive property works both ways

now

binomials you may have done this before

I'm gonna ask you to do it a different

way probably okay because I want you to

do it in a way that makes sense with

what we've been doing so we've been

talking about the distributive property

and when you multiply binomials yes

there are shortcuts but my rule is oh

you always have to understand why before

you take a shortcut okay and I'd take

shortcuts I do

but right now we're not taking you're

shorter okay so best as we look back at

number four I took the a and I multiply

it with the B and with the C that's what

I'm going to do here instead of taking

one term I'm going to take this whole

thing and I'm going to distribute it to

both pieces here so it looks like this

that first piece here is an X and I'm

going to multiply that X with the X plus

three plus the second piece is the one

and I'm going to multiply that with the

X plus three

okay so I need you to see where things

are coming from so this X right here

this plus signs right here this one is

right here okay and so I'm just taking

that X plus three and multiplying it

with the X and taking that X plus 3 and

multiplying it with the one just as we

would with one term you can distribute

more than one term at a time and then we

keep distributing x times X x times 3 1

times X 1 times 3 and then you always

look at the end if you can combine like

terms do it so I have a I have two terms

that have just an X so x squared plus 4x

plus 3 so

you may know how to do this another way

but I'm asking you to do it this way so

when you do homework I want to see it

this way okay and there's a reason for

it when we get to factoring you're gonna

see this done in the reverse order okay

number 10 here's another common mistake

especially coming out of last unit okay

I screw that and I square that and x

squared 2 squared is x squared plus 4 I

seen that a lot

I was wrong very wrong what's wrong

about that it's tempting

okay this is a shortcut that we took

there right let's go that's why

shortcuts can be bad if we don't

understand where shortcuts come from

what does it mean to square an item

means to multiply it with itself so this

means X plus 2 times another X plus 2

right and that's what you're going to do

that

okay and so then it looks like number

line and so what we're going to do is

we're going to take the first X plus 2

and multiply it with that first X and it

looks like this x times X plus 2 plus

take the first X plus 2 and multiply up

to 2 and you get 2 times X plus 2 so

work with the highlighter this X plus 2

here is right here and it's right here

it's been distributed for the other part

and then keep distributing x times X and

x times 2 2 times X and 2 times 2 we get

x squared

plus 4 X plus 4 because what we're doing

now is combining the middle to turn and

that's our answer so try that with 11 do

it the way I've been modeling it please

use the distributive property a couple

times to answer it right now right now

once you do it my way

eventually I'll let you do that no but I

usually take this one and treat both

these so it's going to be X

six

okay so in writing this out you're going

to distribute the X minus six to the X

you write x times X minus 6 and then

it's a plus sign here I'm going to write

plus sign and then I'm going to string

it to the next one

and I go to 6 times another X minus 6 so

the X minus 6 here has been distributed

to the X and it's been distributed to

the slope that's your first step and

then distribute more x squared minus 6x

plus 6x minus 36 combine like terms what

happens to the middle term if you have a

negative

six on a positive six that adds to zero

so this is x squared plus 0x minus 36

but really what's zero times X zero and

so we really don't usually write that

but but know this it is helpful to know

that x squared - 36 is really the same

thing as x squared minus zero X plus

minus or easily that place value is zero

we just usually don't write it but

sometimes we will need to alright the

last part here we're just going to

briefly talk about solving and we'll

keep bringing this theme up but there's

something called the zero product

property that zero product property and

it allows us to use factoring to help us

solve difficult problems and it's a

simple property it says this if you have

two unknowns a and B you multiply them

together and you get an answer equal to

zero that's all you know what can you

tell me about A or B or what can you

tell me at all one of them has to be

zero could they both be zero yes could

they both not be zero No

so one Helly either A or B asked the

Europe and either then would make it

work now zero is the only number that

has this property

you show you I do this a times B equals

3 could you tell me what a and B are

what but it has to be 1 or 3

3:01 does it have to be anyway give me

another combination yeah okay negative 3

and negative 1 or negative 1 in ego 3

anybody give me another option yeah 1.5

times 2 you see what's happening here

why don't you bring in decimals and

fractions there's an infinite number of

possibilities once you use a number

other than 0 but 0 has to be 0 0 is the

only thing that has that property and

for us if we're going to use this

property guess what is the equation has

to be equal to 0 is that the equation is

not equal to 0 we can't use that

property so here's the idea we have

here's our a here's our B a times B

equals 0 so what does that mean well

that means either a 0 or B 0 and so X

could be 0 here or X could be negative 1

and now I have two possible answers for

that problem ok that's how you use that

property so on 13 we don't have

a times B setup here we don't have a

product of two things equal to zero

doing but we can make a product of two

things how by using the distributive

property in factoring what can i factor

out of these okay and what's left if i

divide x squared by X would I get X plus

five equals zero now I have my form a

times B equals zero and so I could split

up my factors and say well either a is

going to be zero or B is going to be

zero if a is zero then this is X is zero

this one would be negative five and now

I have two possible answers and I can go

back and I can plug them into the

original if I plug in negative five up

here at the region I get 25 minus 25

equals zero I plug a zero and I get zero

plus zero equals zero and both work was

it zero product property on the hidden

figures I have to watch it again to see

that but yes there's a lot of math in

that movie so your homework is like this

and and let me show it to you again so

it's the next page one through twenty

four two point one and I'm going to give

you the rest of the time to get started

you