Grade 8 Math Q1 Ep1: Factoring Polynomials
Summary
TLDRIn this educational video, Teacher Joshua introduces viewers to factoring polynomials, focusing on identifying the greatest common monomial factor. Through examples and explanations, he walks students through the process of prime factorization and applying it to algebraic expressions. The lesson aims to enhance critical thinking and mathematical skills, emphasizing the importance of practice and learning from mistakes.
Takeaways
- 📘 The lesson focuses on factoring polynomials, specifically identifying the greatest common monomial factor.
- 🔢 The process of finding the greatest common factor (GCF) involves prime factorization for numbers and considering both numerical and literal coefficients for algebraic expressions.
- 🟠 Prime factorization is illustrated through examples, such as finding the GCF of 8 and 12, which is 4.
- 📚 Variables and their exponents are discussed, emphasizing that a variable raised to an exponent means the variable is multiplied by itself that number of times.
- 🔍 The script provides a method to find the GCF of algebraic expressions by identifying common factors in both numerical and literal coefficients.
- 📐 The term 'polynomial' is defined, and the script explains the components of a polynomial, including terms, variables, constants, numerical coefficients, and literal coefficients.
- ✅ The script demonstrates how to factor polynomials by dividing each term by the GCF and then expressing the polynomial in factored form.
- 📉 Examples are given to practice finding the GCF of various expressions, including cases where the GCF is 1, indicating the expressions are relatively prime.
- 📖 The lesson concludes with a recap of the steps to factor polynomials and a reminder that factoring is the reverse process of polynomial multiplication.
- 👨🏫 Teacher Joshua emphasizes the importance of practice and learning from mistakes to improve mathematical skills and become a better thinker.
Q & A
What is the main topic discussed in the Deaf Ed TV video?
-The main topic discussed in the Deaf Ed TV video is factoring polynomials, specifically focusing on finding the greatest common monomial factor.
What is the first step in finding the greatest common factor (GCF) of two or more numbers?
-The first step in finding the GCF of two or more numbers is prime factorization, where each number is written as a product of prime factors.
How is the greatest common factor of algebraic expressions with variables and exponents determined?
-The greatest common factor of algebraic expressions with variables and exponents is determined by identifying the common variables with the least exponent that appear in each term of the expressions.
What is the greatest common factor of the numbers 8 and 12?
-The greatest common factor of the numbers 8 and 12 is 4, which is the product of their common prime factors 2 times 2.
What is a polynomial according to the Compact Oxford English Dictionary?
-A polynomial is an algebraic expression that shows a sum or difference of two or more terms, containing whole number exponents on the variable.
What is the difference between a variable and a constant in algebra?
-In algebra, a variable is a symbol that represents an unknown value, while a constant is a symbol or number with a fixed value in an algebraic term.
How is the greatest common factor of algebraic expressions with numerical and literal coefficients found?
-The greatest common factor of algebraic expressions with numerical and literal coefficients is found by identifying common factors in both the numerical and literal coefficients and then multiplying them together.
What is a prime polynomial?
-A prime polynomial is one whose greatest common monomial factor is one, meaning it cannot be factored further using the greatest common factor method.
What is the process of factoring a polynomial using its greatest common monomial factor?
-The process of factoring a polynomial using its greatest common monomial factor involves finding the GCF of the numerical coefficients, identifying the variables with the least exponent, multiplying these to get the GCF, dividing the polynomial by this GCF to find the other factor, and writing the polynomial in its factored form.
What is the significance of finding the greatest common factor in mathematics?
-Finding the greatest common factor is significant in mathematics as it helps simplify polynomial expressions, making them easier to work with in calculations and problem-solving.
Outlines
📘 Introduction to Factoring Polynomials
The video begins with an introduction by Teacher Joshua, who welcomes viewers to Deaf Ed TV and announces the lesson's focus on factoring polynomials, specifically the greatest common monomial factor. He encourages viewers to prepare their learning materials and sets the stage for the day's educational journey. The lesson starts with an analogy to commonalities in everyday objects and numbers, leading into the concept of the greatest common factor (GCF) in algebra. Teacher Joshua explains the GCF using prime factorization for numbers and extends this concept to algebraic expressions, illustrating the process with examples involving numbers and variables.
🔢 Understanding Greatest Common Factors in Algebra
This section delves deeper into finding the GCF of algebraic expressions, emphasizing the importance of identifying common numerical and literal coefficients. Teacher Joshua demonstrates how to factor expressions with variables and exponents by breaking them down into their prime factors and identifying commonalities. He uses examples such as 'n cubed' and 'n raised to 5' to show how to determine the GCF when variables are involved. The segment also covers how to handle expressions with numerical and literal coefficients, providing a step-by-step guide on factoring polynomials by extracting the GCF.
📚 Factoring Polynomials Using GCMF
The video continues with a practical approach to factoring polynomials by identifying the greatest common monomial factor (GCMF). Teacher Joshua explains the process of breaking down polynomials into simpler forms by extracting the GCMF. He uses examples to illustrate how to factor binomials and trinomials, emphasizing the importance of dividing each term by the GCMF to find the remaining factors. The segment also touches on the concept of prime polynomials, where the GCMF is one, indicating that the polynomial cannot be factored further.
📐 Practical Examples and Factoring Techniques
This part of the video provides a series of practical examples to reinforce the concept of factoring polynomials. Teacher Joshua guides viewers through the process of finding the GCMF and then factoring the polynomial completely. He also introduces a method of dividing the solution into two parts: identifying the GCMF and finding the other factor. The examples include factoring expressions with multiple variables and terms, demonstrating the application of the concepts discussed in the previous sections.
🏁 Recap and Encouragement for Continued Learning
In the final segment, Teacher Joshua recaps the key points of the lesson, emphasizing the importance of understanding the process of factoring polynomials and the concept of the GCMF. He encourages viewers to practice and apply these skills, acknowledging that making mistakes is a part of learning. The video concludes with a reminder of the importance of math in developing critical and logical thinking skills and a prompt for viewers to continue their learning journey with the next topic on special products in factoring polynomials.
Mindmap
Keywords
💡Factoring Polynomials
💡Greatest Common Monomial Factor (GCMF)
💡Prime Factorization
💡Polynomial
💡Variable
💡Exponent
💡Coefficient
💡Relatively Prime
💡Monomial
💡Factored Form
Highlights
Introduction to factoring polynomials, specifically focusing on the greatest common monomial factor.
Observation exercise involving images to identify commonalities, setting the stage for understanding common factors.
Explanation of how to find the greatest common factor (GCF) using prime factorization for numbers.
Demonstration of finding the GCF of algebraic expressions with variables and exponents.
Definition and explanation of key terms such as polynomial, variable, constant, numerical coefficient, and literal coefficient.
Tutorial on determining the GCF of algebraic expressions with numerical and literal coefficients.
Methodology for finding the GCF of monomials that do not initially appear to have common factors.
Discussion on the concept of relatively prime monomials and their GCF being one.
Step-by-step guide on factoring polynomials by extracting the greatest common monomial factor.
Example exercise showing how to factor a binomial by identifying the GCF and simplifying.
Explanation of the difference between finding the GCF and writing a polynomial in its factored form.
Practical example of factoring a polynomial with multiple terms and variables.
Clarification on the term 'prime polynomials' and their characteristic of having a GCF of one.
Recap of the steps involved in factoring polynomials using the greatest common monomial factor.
Interactive quiz to test understanding of factoring polynomials with immediate feedback.
Encouragement for learners to practice at home with additional exercises for reinforcement.
Final thoughts from Teacher Joshua on the importance of learning from mistakes and continuous improvement in mathematics.
Closing remarks and a preview of the next topic: factoring polynomials known as special products.
Transcripts
[Music]
[Music]
good day everyone
and welcome to deaf ed tv i am teacher
joshua
and i will be your guide in sharpening
your scales and enhancing your minds in
order to face the challenges
here in grade 8 mathematics ready your
self-learning mojos
your paper and your pen with you let us
have
a wonderful day of learning
the first lesson that we will tackle is
factoring polynomials
specifically the greatest common
monomial factor
but before that i would like you to
observe the following images
can you give a description that is
common to all of them
they are all orange in color what else
they're all good for your health how
about this
what is common among these four
even if they are of different colors we
know that all of these are crayons
what if i ask you to think what is
common with 2
4 and 6 they are all
even numbers or divisible by 2.
if we look at their factors two has
one and two four has one
two and four six has
one two three and six
observe that aside from one all of them
has a factor of 2. how about on this
expressions x 2x
3x and 4x they all have the same
variable x like in english
in science we can also find
commonalities in algebra
specifically the greatest common factor
it is just what you did in the constants
2 4 and 6
and expressions x 2x
3x and 4x easy right
in this lesson we will factor
polynomials
with a common monomial factor let us
start with a review
on how to get the greatest common factor
and
one way to find the greatest common
factor
of two or more numbers is prime
factorization
let us determine the greatest common
factor
of 8 and 12. first thing to do is
write each number as a product of prime
factors eight is equal to the product of
two
times two times two well twelve is equal
to two times two times 3 do you see
the same number in each set of factors
next is we identify common prime factors
of the given numbers we can show them by
encircling
the common factors by pairs last
is to multiply the common factors so
what
is the greatest common factor of 8 and
12
the greatest common factor is 4 which is
the product
of the common factors 2 times 2.
how about the greatest common factor of
algebraic expressions
where there are variables and exponents
since a variable represents
an unknown value we cannot use prime
factorization
to find their greatest common factor so
how can we find the gcf
of n cubed n raised to 5 n raised to 6
and n raised to 9. what do you remember
about a variable n
raised to an exponent it means that
n raised to an exponent k means
n is multiplied to itself k times
based on the given exponent so n cubed
is
n times n times n same goes to the
expression
n raised to five it is equivalent to the
product of
n written five times also with the other
expressions
what did you observe how many n is
common in the factors
the greatest common factor is n cubed or
n raised to 3 from n times n times n
let us review the following terms
according to the compact
oxford english dictionary the word
polynomial
comes from the latin greek words
poly meaning many and nomen
which means name or term therefore
polynomial means many terms a polynomial
is an algebraic expression that shows a
sum
or difference of two or more terms
containing
whole number exponents on the variable a
variable
is a symbol which represents an unknown
value
a constant is a symbol or a number with
fixed
value in an algebraic term numbers form
the numerical coefficient when symbols
form the literal coefficient
in this example the expression is
4 x squared minus 2. this
is a polynomial since the variables have
non-negative
whole number exponents this polynomial
has two terms
four x squared and two the first term
has a numerical coefficient four and a
literal coefficient
x squared an exponent refers to the
number of times
a number is multiplied by itself the
last term
is a constant term since there are no
variables
multiplied to the number
now we will determine the greatest
common factor of
algebraic expressions with numerical and
literal coefficients
find the greatest common factor of four
x cubed
and eight x squared four x
cubed can be expressed as a product of 2
times 2 times x times x
times x and 8 s squared
is equal to 2 times 2 times 2
times x times x next step
is we identify common factors we can
separate
and align the numerical and literal
coefficients
to make this easy can you identify which
is common
among the prime factors how many two are
common
how many x are common we can encircle
these pairs to help us determine the
common factors
the common factors are 2 2
x and x last is to get the product of
these common factors
the product is the greatest common
factor of the given
four x cubed and eight x squared we will
multiply
two times two times x times x
the greatest common factor of four x
cubed
and eight x squared is four x squared
[Music]
now let us try this example find the
greatest common factor
of these expressions 15 y raised to the
power of 6
and 9z the first step is to enumerate
the prime factors what is the prime
factorization
of 15y raised to 6. the numerical
coefficient
can be factored as 3 times 5 while
y raised to 6 means that y
is multiplied to itself 6 times
how about 9z 9z is equal to 3
times 3 times z can you identify
all common factors note that 3 is the
only
common factor hence the greatest common
factor
of 15y raised to the power of 6
and 9z is 3.
that is one method we can use to
determine the greatest common factor
of polynomials but there are some cases
where it seems that the given pair of
monomials
do not have a common factor observe
these monomials
4 a squared and 9 b squared
let us start with the numerical
coefficients four
and nine four can be expressed as the
product
two times two and nine can be expressed
as three times three do they have common
factors
in a glance they seem to have no common
factors
but remember that one is a factor of any
number
hence the greatest common factors of
four
and 9 is 1. next
are there any common factors for the
literal coefficients
a squared and b squared since the
expressions
do not have common variables we can
conclude that 1 is the greatest common
factor of the given monomials
therefore the greatest common monomial
factor
of 4a squared and 9b squared
is one
and if the gcf of the expressions is one
we can say that they are relatively
prime
how about if we get the greatest common
factor
of three monomials what is the greatest
common factor
of six a squared b squared three a
b squared and fifteen a cubed
b squared first get their prime factors
six a squared b squared is equal to two
times three times a times a times b
times b how about three a b squared
three
a b squared is just three times a
times b times b note that
we use two variables here so it can be
helpful to separate them
in our solution then we have 15
a cubed b squared this
monomial can be written as 3 times 5
multiplied with a times a times a
multiplied with b times b because of the
indicated
exponents now identify the common
factors
we will have these numbers and variables
and lastly we write the product of these
factors
what would be our final answer the
greatest
common factor of the expressions 6
a squared b squared 3 a b squared
and 15 a cubed b squared is the product
3 times a times b times b
which is three a b squared
[Music]
we've exercised our eyes on finding
common characteristics of things and we
exercise
our minds by reviewing prime
factorization and the greatest common
factor
we may now proceed on how we can use
these skills
let us have an exercise in getting the
greatest common factor of these
expressions
let us start with the greatest common
factor of 12a
and 18 abc the greatest
common factor of 12a and 18
abc is 6a next what is the greatest
common factor
of 4x 12x squared and
8 the answer is 4.
how about the greatest common factor of
6x
14y and 15z
since all three expressions do not have
a common prime factor
we can say that they are relatively
prime hence
the greatest common factor of 6x
14y and 15z is 1.
notice that in the previous examples
prime factorization
is used to find the greatest common
factor
of the given pair of monomials the next
examples
illustrate how to factor polynomials by
getting the greatest
common monomial factor let us write the
expression
6x plus 3x squared in factored form
we can factor this polynomial by getting
the greatest common factor
of each term and i will introduce a new
method
to acquire the common factors we can
divide the solution into two parts
one for identifying the greatest common
factor
of the polynomial and then finding the
other factor
with the help of the identified
expression 6x
plus 3x squared is a binomial 6x
is the first term and 3 x squared is the
second term
the numerical coefficient of 6x in the
first term
is 6 and 3 in the second term
observe the literal coefficients the
simplest way to identify
their greatest common factor is to get
the common variables with the least
exponent by prime factorization we will
get a greatest common factor
3x to factor 6x
plus 3x squared simply divide each term
of the given polynomial by the common
factor
6x divided by 3x and 3x squared
divided by 3x by applying division and
the quotient rule in the loss of
exponent
6x divided by 3x is equal to 2
since x divided by x is just 1.
then what is 3x squared divided by 3x
3 divided by 3 is 1 and x squared
divided by x
is just x do not forget the signs to
complete
our polynomial factor last step is to
write the polynomial
in factored form the polynomial 6x plus
3x
squared can be written in factored form
as 3x
multiplied by the quantity 2 plus x
remember that getting the greatest
common monomial factor
is different from writing a polynomial
in its factored form
factoring is often called the reverse
process
of multiplying polynomials where we
write a polynomial
as a product of two or more simpler
polynomials
now let us try this example right 12x
cubed
y raised to 5 minus 20
x raised to 5 y squared z in complete
factored form again we can break it down
into parts
so we can focus on each step while
factoring
first find the greatest common monomial
factor
12 is equal to 2 times 2 times 3
and 20 is equal to 2 times 2
times 5. then we get the common
variables in each term
which has the least exponent so what is
the greatest common monomial factor
it is 4 x cubed y squared
then find the other factor by dividing
each term of the polynomial by the
greatest common monomial factor
the operation can be shown as such we
divide
12 by 4 and subtract exponents of
expressions
with similar bases the first term of the
factor
will be 3y cubed minus the second term
what do you think will it be 20 divided
by 4
is 5 then applying the loss of exponents
we will have x squared and z
since y squared divided by y squared is
1.
step 5 write the polynomial in factored
form
in this example we have shown that 12
x cubed y raised to 5 minus
20 x raised to 5 y squared
z is equivalent to 4 x cubed
y squared times the quantity three y
cubed minus five x squared z
[Music]
let us try the next example get the
factors
of two x plus three y squared what
is the greatest common factor of two and
three
since these numerical coefficients and
constant
are relatively prime the greatest common
factor
is one
how about the literal coefficients they
do not have
any common variables hence the greatest
common factor
is also one
similar to our previous example in
finding the greatest common monomial
factor
there will be instances that it is one
we call these polynomials
prime polynomials and there you have it
we have discussed how to factor
polynomials by getting the greatest
common monomial factor
let us have a recap one factoring is the
reverse process
of multiplication where we write a
polynomial
as a product of two or more simpler
polynomials
two the greatest common monomial factor
of two or more expressions is the
product
of the gcf of the numerical and literal
coefficients three prime polynomials
are polynomials whose greatest common
monomial factor
is one and can only be written as a
product of one
and the polynomial itself number four to
factor a polynomial using its greatest
common monomial factor
we can consider these steps a
find the greatest common monomial factor
of the numerical coefficients
b find the variables with the least
exponent
that appears in each term of the
polynomial
it serves as the gcf of the literal
coefficients
c get the product of the greatest common
factor
of the numerical coefficient and the
variables
the product serves as the greatest
common monomial factor
of the given polynomial b
find the other factor by dividing the
given polynomial
by its greatest common monomial factor
and e write the final factored form
of the polynomial since we are near the
end of your lesson
prepare your pens and your paper because
it is
important to evaluate what you have
learned i will give you 10 seconds to
answer
each item
[Music]
write a polynomial factor in the blank
to complete
each statement number one seven p
squared minus seven p is equal to
seven p times the quantity of what
polynomial
the answer is p minus one
number two 18 x y plus three y
is equal to blank times the quantity of
six x
plus one
the polynomial 18xy plus 3y
can be written as the product 3y times
the binomial
6x plus 1. number three
15 m cubed minus 15
m squared plus 20 m is equal to 5
m times what polynomial
the polynomial factor in this item is
three m squared
minus three m plus four
number four seventeen x raised to five
minus fifty one x raised to four minus
thirty four
x in factored form is blank times the
quantity
of x raised to four minus three x
cubed minus 2
[Music]
the greatest common monomial factor is
17x
number five what is the factor of the
polynomial
35 x raised to 5 y squared
plus 21 x raised to 4 y
plus 14 x cubed y squared if the
greatest common factor
is equal to 7 x cubed y
the answer is five x squared y plus
three
x plus two y how are you in the short
seat work
did you get a high score nevertheless i
know you have done a great job remember
that it
is okay to commit mistakes at the start
the important thing is you learn from
these mistakes
and apply that knowledge to become
better in math
and in life as an additional exercise in
practice at home
try answering the activity on the self
learning module
1a at page 10 activity 1
break the grade
i hope that you have learned a lot in
our episode today
note that there are many ways to factor
polynomials and you must focus on the
key
concepts and process in factoring these
polynomials
with a little more practice i believe
that you can
ace any lesson in mathematics
next topic will be factoring polynomials
that we call
special products remember
math is not only about numbers and
operations it
is an exercise for our minds for us to
be critical
logical and responsible thinkers
again this is teacher joshua reminding
you to keep safe
have a nice day and see you next time
[Music]
you
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