The Distributive Property (M2 2.1 Notes)
Summary
TLDRThis 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
- đ The lesson focuses on polynomials, specifically on multiplying and factoring them, with an emphasis on the distributive property.
- đą Adding and subtracting like terms is covered, where coefficients are combined without changing exponents.
- 𧩠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.
- đ The distributive property is explained as a method to interact between multiplication and addition, and it is also used in reverse for factoring.
- đą When multiplying terms, exponents change, which is different from addition or subtraction where they remain the same.
- đ Factoring is demonstrated by finding the greatest common factor and dividing each term by it, which can be checked by multiplying the factors.
- đ 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.
- đ« The importance of understanding the underlying principles before using shortcuts is stressed to avoid common mistakes.
- đ The process of multiplying binomials is detailed, emphasizing the step-by-step application of the distributive property rather than relying on shortcuts.
- â 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
đ 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.
đ 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.
đ 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.
đ 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.
𧩠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.
Mindmap
Keywords
đĄDistributive Property
đĄCombining Like Terms
đĄFactoring
đĄExponents
đĄZero Product Property
đĄParentheses
đĄGreatest Common Factor
đĄBinomials
đĄSolving Equations
đĄLike Terms
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
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