Grade 8 Math Q1 Ep1: Factoring Polynomials

Summary

In 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

00:00

📘 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.

05:02

🔢 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.

10:05

📚 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.

15:05

📐 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.

20:07

🏁 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.

Keywords

💡Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or factors. In the video, this concept is central to the theme as it is the main mathematical operation being taught. The script provides examples such as factoring out the greatest common monomial factor from expressions like '6x + 3x^2' which can be factored into '3x(2 + x)'.

💡Greatest Common Monomial Factor (GCMF)

The Greatest Common Monomial Factor refers to the highest degree of a monomial that is a factor of each term in a polynomial. In the script, the GCMF is identified by examining both numerical and literal coefficients. For instance, in the polynomial '4x^3' and '8x^2', the GCMF is '4x^2', as it is the product of the common numerical coefficient '4' and the least power of 'x' present in both terms.

💡Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. In the video, prime factorization is used to find the GCMF of numerical coefficients. For example, to find the GCMF of 8 and 12, the script demonstrates breaking down each number into its prime factors: 8 = 2 x 2 x 2 and 12 = 2 x 2 x 3, from which the GCMF is determined to be 4.

💡Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. The script explains that polynomials can be factored, and it uses terms like '4x^2 - 2' to exemplify a polynomial with two terms.

💡Variable

In algebra, a variable represents an unknown value that can be used in operations and equations. The script mentions variables when discussing the GCMF of algebraic expressions, noting that variables raised to an exponent indicate repeated multiplication.

💡Exponent

An exponent indicates the number of times a base number is multiplied by itself. In the script, exponents are used to describe the power to which a variable is raised, such as in 'n^3' which means 'n' multiplied by itself three times.

💡Coefficient

A coefficient is a numerical factor in a term of a polynomial, which multiplies the variable. The script differentiates between numerical and literal coefficients, explaining how to factor out the GCMF by considering both types of coefficients.

💡Relatively Prime

Numbers or expressions are relatively prime if they have no common factors other than 1. In the video, the script uses this term to describe monomials that do not share any common factors, such as '4a^2' and '9b^2', which are relatively prime because they involve different variables.

💡Monomial

A monomial is a single-term expression consisting of a product of numbers and variables. The script discusses monomials in the context of finding the GCMF, noting that the GCMF of a set of monomials is the product of the common numerical and literal factors.

💡Factored Form

Factored form is a way of expressing a polynomial as a product of its factors. The script demonstrates converting polynomials into factored form by identifying and extracting the GCMF, such as expressing '6x + 3x^2' as '3x(2 + x)'.

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