Factoring Polynomials using Greatest Common Monomial Factor

Summary

This educational video script introduces grade 8 students to the concept of factoring polynomials using the greatest common monomial factor (GCMF). It explains the process of identifying factors and GCMF through examples with numbers and algebraic expressions. The script also covers prime factorization and demonstrates how to apply these techniques to factor binomials and polynomials, including the difference of cubes and sum of cubes, with the aim of simplifying complex expressions for easier understanding.

Takeaways

• 📚 The lesson is focused on factoring polynomials using the greatest common minimal factor, also known as the greatest common factor (GCF) or greatest common monomial factor (GCMF).
• 🔢 A factor is defined as a number or algebraic expression that divides another number or expression evenly with no remainder.
• 🌰 Examples are given to illustrate how to find the factors of numbers such as 20 and 10, and then identify their greatest common factor (GCF).
• 🔍 The script introduces the concept of prime factorization as a method to find the GCF of algebraic expressions, which involves breaking down numbers into their prime factors.
• 📝 The greatest common monomial factor (GCMF) is explained as the common factor with the smallest exponent for variables in algebraic expressions.
• ✂️ The process of dividing each term of a polynomial by the GCMF to find the other factors is described, using examples such as 6x + 3x^2 and 12x^3y^5 - 20x^5y^2z.
• 📉 The importance of identifying common factors in polynomials is emphasized to simplify the expression and make factoring easier.
• 📚 The script provides a step-by-step approach to factoring polynomials, including dividing each term by the GCMF and simplifying the result.
• 📈 The concept of exponents is applied when finding GCMF, where the variable with the smallest exponent is chosen for the factor.
• 📝 The script also covers how to handle terms without common variables or coefficients in the process of finding the GCMF.
• 🔚 The lesson concludes with a reminder to check the factored form by multiplying the factors to see if they yield the original polynomial, and a teaser for upcoming lessons on additional factoring techniques.

Q & A

What is the definition of a factor in mathematics?

A factor is a number or algebraic expression that divides another number or expression evenly with no remainder.

What are the factors of 20?

The factors of 20 are 1, 2, 4, 5, 10, and 20.

How do you find the greatest common factor (GCF) of two numbers?

You list all the factors of each number and find the greatest factor that is common to both numbers.

What is the GCF of 20 and 10?

The GCF of 20 and 10 is 10.

What is the greatest common monomial factor (GCMF) and when is it used?

The GCMF is the greatest common factor of algebraic expressions or monomials, and it is used when dealing with polynomials or expressions that contain variables.

How does the method of prime factorization help in finding the GCMF of algebraic expressions?

Prime factorization breaks down the numbers into their prime factors, making it easier to identify and align the common factors in the expressions.

What is the GCMF of 4x^3 and 8x^2?

The GCMF of 4x^3 and 8x^2 is 4x^2.

How do you factor a polynomial using the GCMF method?

You find the GCMF of the terms in the polynomial, divide each term by the GCMF, and then multiply the GCMF by the resulting factors.

What is the factored form of 6x + 3x^2 using the GCMF method?

The factored form of 6x + 3x^2 is 3x(2 + x).

What is the GCMF of 12x^3y^5 - 20x^5y^2z and how is it used to factor the expression?

The GCMF of 12x^3y^5 - 20x^5y^2z is 4x^3y^2. It is used to divide each term and find the remaining factors, resulting in the factored form 4x^3y^2(3y^3 - 5x^2z).

How does the script differentiate between the GCF and GCMF?

The script differentiates by using the term GCF for numbers and GCMF for algebraic expressions or monomials, emphasizing the variable with the smallest exponent in the GCMF.

Outlines

00:00

📚 Introduction to Factoring Polynomials

This paragraph introduces the concept of factoring polynomials for 8th-grade mathematics. The focus is on using the greatest common minimal factor to factor polynomials. It starts by defining a factor as a number or algebraic expression that divides another evenly with no remainder. Examples given include finding the factors of 20 and 10, and then determining the greatest common factor (GCF) of these numbers. The GCF is identified as the largest number that divides both numbers without a remainder. The paragraph also introduces the concept of the greatest common monomial factor (GCMF) for algebraic expressions, which is similar to GCF but applies to variables and monomials.

05:02

🔍 Prime Factorization and GCMF

The second paragraph delves into the method of prime factorization to find the greatest common monomial factor (GCMF) of algebraic expressions. It explains that prime numbers, which have only two factors (one and itself), are used in this method. The process involves breaking down expressions into their prime factors and then identifying the common factors between two or more expressions. Examples are given to demonstrate how to factor expressions such as 4x^3 and 8x^2, and 15y^6 and 9z, by finding their GCMF and then dividing each term by this factor to simplify the expression.

10:05

📘 Factoring Polynomials Using GCMF

This paragraph continues the discussion on factoring polynomials by applying the concept of the greatest common monomial factor (GCMF). It provides a step-by-step approach to factor binomial expressions like 6x + 3x^2 by identifying the GCMF from both the numerical coefficients and the variables, focusing on the variable with the smallest exponent. The method involves dividing each term of the polynomial by the GCMF and then simplifying to find the remaining factors. An example is given to illustrate the process, which results in the factored form of the polynomial.

15:06

📙 Advanced Factoring with GCMF

The fourth paragraph presents a more complex example of factoring polynomials using the greatest common monomial factor (GCMF). It involves a trinomial expression, 12x^3y^5 - 20x^5y^2z, and demonstrates how to find the GCMF by examining both the numerical coefficients and the variables. The process includes dividing each term by the GCMF and simplifying to find the remaining factors. The example concludes with the factored form of the polynomial, showcasing the application of the GCMF in a more advanced context.

20:06

📕 Factoring Trinomials and Beyond

This paragraph extends the factoring technique to trinomial expressions, specifically 28x^3z^2 - 14x^2y^3 + 36yz^4. It discusses the process of identifying the GCMF among the terms, which involves examining both the numerical coefficients and the variables. The paragraph emphasizes the importance of recognizing which variables are common to all terms and which are not. After determining the GCMF, the method involves dividing each term by this factor and simplifying to find the remaining factors, leading to the factored form of the trinomial.

25:08

📓 Factoring with Non-Perfect Cubes

The final paragraph addresses the factoring of expressions that do not represent perfect cubes, such as 5h + 40hk^3. It clarifies that these expressions cannot be factored as sums of cubes and instead requires finding the greatest common monomial factor (GCMF). The process involves dividing each term by the GCMF, which in this case is 5h, and then simplifying to find the remaining factors. The paragraph concludes with an example that demonstrates this factoring method and hints at further lessons on factoring techniques.

Keywords

💡Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of its factors. In the video, this concept is the central theme, as the instructor discusses various methods to find the greatest common factor of polynomials, which is essential for simplifying and solving equations. For example, the instructor explains how to factor expressions like '6x + 3x^2' by identifying the greatest common monomial factor, which in this case is '3x'.

💡Greatest Common Factor (GCF)

The GCF is the largest number that divides two or more integers without leaving a remainder. In the context of the video, the GCF concept is extended to algebraic expressions, where the instructor explains how to find the GCF of numbers like 20 and 10, which is 10. This concept is foundational for understanding how to simplify and factor expressions in mathematics.

💡Monomial

A monomial is a single-term algebraic expression. In the video, the instructor introduces the concept of the 'Greatest Common Monomial Factor' (GCMF), which is similar to the GCF but applies to monomials. For instance, when factoring '4x^3' and '8x^2', the GCMF is '4x^2', as it is the largest monomial that divides both expressions.

💡Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors, which are prime numbers that multiply together to result in the original number. In the video, the instructor uses prime factorization to find the GCF of expressions, such as breaking down 20 into 2 x 2 x 5, and using these factors to identify the GCF of algebraic expressions.

💡Variable

In algebra, a variable represents an unknown value. The video script discusses variables in the context of factoring polynomials, emphasizing that when identifying the GCMF, one should consider the variable with the smallest exponent. For example, between 'x^3' and 'x^2', 'x' is the variable with the smallest exponent and thus part of the GCMF.

💡Exponent

An exponent indicates the number of times a base is multiplied by itself. In the video, the instructor explains how to handle exponents when finding the GCMF, such as subtracting the exponents when dividing terms like 'x^5' by 'x^2', resulting in 'x^3'.

💡Binomial

A binomial is a polynomial with two terms. The video script uses binomials to demonstrate the process of factoring by finding the GCMF. For example, '6x + 3x^2' is a binomial, and the GCMF is '3x', which is then used to factor the expression.

💡Tricomial

A trinomial is a polynomial with three terms. The video script extends the concept of factoring to trinomials, such as '28x^3z^3 - 14x^2y^3 + 36yz^4', where the GCMF is determined to be '2', and the expression is factored accordingly.

💡Sum and Difference of Cubes

This is a specific factoring technique used for expressions that are sums or differences of two cubes, such as 'a^3 + b^3' or 'a^3 - b^3'. Although not explicitly detailed in the script, the instructor hints at this technique, which is relevant when discussing factoring polynomials.

💡Perfect Cube

A perfect cube is a number that can be expressed as the cube of an integer. In the video, the instructor checks if terms like '5h' and '40h' are perfect cubes before determining the GCMF. The concept is important for recognizing when a polynomial can be factored using specific formulas for cubes.

Highlights

Introduction to the concept of factors in mathematics, defining a factor as a number or algebraic expression that divides another evenly with no remainder.

Listing method for finding factors of numbers, demonstrated with examples of numbers 20 and 10.

Explanation of the greatest common factor (GCF) and its significance in mathematics.

Identification of the GCF for 20 and 10 as 10, showcasing the process of elimination to find the greatest common factor.

Introduction of the term 'Greatest Common Monomial Factor' (GCMF) for algebraic expressions.

Transition from GCF to GCMF when dealing with algebraic expressions, using the example of 4x^3 and 8x^2.

Prime factorization method for finding GCF, explained with the example of 4x^3 and 8x^2.

Finding the GCMF of polynomials using prime factorization, demonstrated with 15y^6 and 9z.

Method for factoring polynomials using GCMF, illustrated with the binomial 6x + 3x^2.

Tip on selecting the variable with the smallest exponent when finding GCMF for variables.

Example of factoring a polynomial with multiple terms, 12x^3y^5 - 20x^5y^2z.

Process of dividing each term by the GCMF to find the other factors in a polynomial.

Explanation of how to handle terms without a common variable or exponent when factoring.

Factoring a trinomial polynomial, 28x^3z^2 - 14x^2y^3 + 36yz^4, using GCMF.

Illustration of the prime factorization for numbers in a polynomial to find the GCMF.

Final example of factoring a binomial with terms 5h and 40hk^3, identifying the GCMF as 5h.

Factoring the remaining terms after identifying the GCMF, using the example of 1 + 8k^3.

Conclusion of the lesson with a teaser for upcoming lessons on additional factoring techniques.

Transcripts

good day everyone

and today is the first day when you're

going to

answer your module so

the first topic for grade 8 mathematics

is about factoring polynomials so this

time

i'm going to discuss to you how to

factor polynomials

using greatest common minimal factor

but let us recall first what

is factor

what do you mean by factor when you say

factor

that is a number or algebra expression

that divides another number or

expressions

evenly that is with no remainder

okay say for example 20

so we're going to search for numbers

for a number or numbers that divides the

another number

or expressions or these 20 evenly with

no remainder

okay so what are those so

let's list all the factors of

20

okay of course one

why one because since we when you're

going to divide 20 by 1

that is 20 so there is no remainder

how about 2 okay 2. so when you divide

20 by 2 that is 10 so

still 2 is a factor of twenty

okay next four why

since twenty divided by four that is

five that is

also a whole number no remainder

next five okay when you

when you divide 20 by 5 that is

4 and x 10

so 20 divided by 10 is 2. so there is no

remainder again

and last one is 20. so the twenty

divided by twenty

that is one so the factors of

twenty are one two

four five ten and twenty

let's have another example how about

10 let's find

the factors of 10

okay so again factors is a number or a

expression that divides another number

or expressions even liters

with no remainder of course

one so okay um one

is a factor to any expression or to any

number actually class

okay two yes two ten divided by two that

is five

five ten divided by 5 is 2

there's no remainder and lastly 10.

the factors of 10 are 1 2 5

10. so this method is listing method

release

all the factors okay

from these two numbers 20 and 10 we're

going to look at the

common factor that is the greatest

common factor

i guess everybody is very

they know or you know what is greatest

common factor

greatest common factor so look for

the number the co the common factor is

the greatest

so here so one is

the common factor for 20 and 10 2

five and one two five

and ten that is their common

they are the common factors but we're

going to search

for the greatest so obviously

the gcf this common factor is everywhere

it's ever basic evasion is gcf

the gcf of 20 and 10

is 10 okay

that's it 20 20 and fourteen

and ten is ten

okay again the gcf of twenty and 10

is 10. okay how about

if we're going to get

the gcf of 4x cubed

and eight x squared

so as you notice there uh there is x

cubed and x squared so we're not dealing

here purely numbers

so we're dealing here algebraic

expressions

or variables so since we're dealing with

that algebra expressions

so we're not going to use the word gcf

instead we're going to use g c

and f k g

g that is greatest c

for common m for monomial

f that is factor so m

is just the new word here so dcmf is the

greatest

common monomial factor okay

a while ago we use listing method in

finding the gcf of 20 and 10.

here we can do that also but

we're where it's really time consuming

we're running out of time or we're going

to consume

um more time so i'm going to introduce

to you

a new or

this is not you this was introduced to

you when you were elementary

new this method this is

prime prime factorization

okay from the word prime so

we're going to look factors that are

prime

or prime numbers so what is this prime

number

so to recall prime numbers are numbers

whose factors are only one and itself

example two two factors of two are only

two and one so two on itself two and

no one and itself so one and two three

three the factors of three are only

three and one

okay let's do that

four x cubed

okay four look for the prime factors

that would be

two times two x cubed

that would be x times x

times x so x is multiplied three times

so that would be that would result to x

cube

next eight x squared

so eight that is three two so two

times two okay there is times two

but in prime factorization the tip class

you need to align those factors that are

similar

so since here it's x here so it's not

this that's not the same with two or

different from two

so you write another two here so times

okay how about x squared

x squared is just times x so you're

right here

times x okay

and then you look for their

common factors so here too

okay and now i crush it so that

it's simple it's it's it's

more simple for us to determine

or to crash out the used common factor

okay another

common another two times

another common x another common

x but here it is x

is no common with x eight x squared

this also two has no common with 4 x

cube

and then so you multiply

common factors common prime factors

2 times 2 that is 4

x times x is x squared

okay so meaning

the g c m f of 4 x

cubed and 8 x squared is

4 x squared another example in finding

the gcmf or the greatest common factor

k let's have here 15

y to the power of 6 and

i'm sorry we're not going to write 9

and 9 z okay

sub it so we're going to use the prime

factorization method since this thing

method is really hard

so severe 15

15 y the power six

so 15 that is so find the prime factors

is 3 times y both 3 and 5 are prime

numbers

3 times 5 and y divided by 6 is

y multiplied six times so

y times y times

y times y times

y times y okay

that's it how about for nine z

okay nine z

okay four nine oh that's obviously

the prime factors is r

nine three multiplied by another three

so that's three since

after three four fifteen white barbie

six is five

so you will put your three and then for

z

there is no zero since they're y's

so multiply by z now look for

look for their common factor

so it's very clear

the common factor is only three so

meaning

the gcm f of 15 y to the power

6 and 9 z is

three okay we're done finding the gcf or

the gcm f

of two terms or two monomials now let's

have

polynomials so you're going to find the

gcmf

so let's have your number one

six x plus

three x squared

so as you notice there are two terms

here this this is binomial

so what we're going to do is you look

for their

of course gcmf so you have

let's start with the number

six and three

they're common or they're gcf or i mean

yeah you see f

is three okay

for x and x squared their gcm f

is also x

tips or um yeah i'll give you a tip

so when you're going when you're given

with variables

the gcf there is the variable with the

smallest exponent

for so for x and x squared that's x

okay and then you look for

the other factor so 3x is obvious

obviously the gcmf or the greatest

common monomial factor

but we're going to look also for the

other factor

how so you just divide

each term so yeah there is 6x

you divide it by the gcmf which is the

3x plus 3

x squared divided by

the gcm x also this 3x

okay next

copy 3x

so now you have where 6x divided by 3x

so 6 divided by three that is two

x divided by x that is

one okay

way one okay for example five divided by

five

you're not going to say zero five

divided by five

is one so if you have x divided by x

that is one

so two times one that's obviously

that's obviously 2. next

3 x squared divided by 3 x

3 divided by 3 that is 1 also

x squared divided by x

squared i am in x so that is x

y what i did here is

just subtract the exponent k

recall right here i recall

and i know rules and i mean loss and

of exponent so if you have your a

8 the power of m over or divided by

a to the power of n so what

are you going to do here is just you

just subtract the exponent so

m minus n okay

recall so here x squared

so 2 for the x class there is no

exponent

that you see for x but obviously the

exponent

for x is one understood nanga

one an exponent so two minus one that is

one or it's just

x why is it i did not put one

for the numerical coefficient of x

of course it's obvious the numerical

coefficient of x

is one no need to write it

okay so again

the factors of six

x plus three

x squared are three x

plus i mean multiplied

multiplied by the quantity of two

plus x okay

class you can check actually your

answers

okay i'll give you a tip here how

so three x you just multiply

the given factors or the factors that

you

that you have that you got

okay you multiply so here you distribute

3x times 2 that is of course

6x 3x times

x that is 3x squared

so what did you know not notice

so you have you come up with the product

of the given polynomial

let's have another example in factoring

factoring polynomials using

gcmf okay so you have your number two

12x cubed y to the power of 5

minus 20 x to the power of 5

y squared z okay

so you copy 12 x cubed

y to the power of 5 minus 20

x to the power of 5 y squared

z okay so you look for the gcmf so let's

start with the numbers

so for 12

so for 12 and 20

obviously their gcmf is

yes four

okay four since when you divide twelve

by four that's three twenty divided four

is five so

that's a whole number it's no remainder

okay for

x cubed and x to the power of five

okay x cubed and x five so

i said it a while ago that the tip

for finding the gcf or gcmf

is to select the variable was the

smallest

exponent so that is

that is x cube

okay x cube now for y

to power 5 and y squared so that's

obviously

y squared okay how about z

as you notice a 12

x cubed y by 5 there is no z so meaning

that's obvious that z

is the not common factor for the two

terms

okay so meaning 4 x cubed y

squared is the gcm f

now let's look for the other factor

how so just divide each term by the gcm

f

4x cubed y squared so you have 12

x cubed y

to the power of 5 divided

by 4 x cubed

y squared minus

20 x to the power of 5

y squared z

divided also with the gcm f 4

x cubed y squared

okay okay let's do it

four x cubed y

squared okay twelve divided by four

that's three

x cube divided by x cubed that's one one

times three that's three only

y to the power five divided by y squared

as i said just subtract the exponent

five minus two that is

three so three y cubed

minus

okay minus 20 divided by four

that is five

okay x to the power of 5 divided by x

cubed

so just subtract the exponent that would

be

5 minus 3 that is squared

okay for y squared divided by y squared

is 1

multiplied by 5x squared obviously this

5x squared

then z since there is no z in the

denominator so just write it

okay so meaning

the factors for 12 x cubed y

to the power 5 minus 20

x to the power of 5

y squared z are

4 x squared i am in cube

y squared multiplied by three

y cubed minus

five x squared

z class

remember i told you you can check your

answers by multiplying the factors

and then you come up with the given

polynomial

okay another example so you have your 28

x cubed z squared minus

14 x

squared y cubed plus

36 y z to the power of four

so you have your three terms it's

trinomial

okay so just let's copy first

28

x cubed z cubed

minus 14 x squared

y cubed plus 36

y z to the power of four

so the same thing you're going to find

our gcm f

so let's start with the numbers 28 14

36 so for this class when you're going

to use the listing method or the

prime factorization you come up with

they just the gcmf of

okay 36 is 4 times nine fourteen is

seven times two twenty eight is

um seven is four so the

common is two

okay okay

okay i'll illustrate you birds

it's if you can

um visualize it okay let's have beer

28 14

36 so 28 prime factorization tile

fourth that's four times seven so four

is it's not prime so it's two

times two times seven

fourteen is two times seven

thirty-six is four times nine

yes nine four is two

times two nine is

the three times three so very clear

they're common factors form factor

it's just two that is why i write a well

ago

two okay

okay so there you see

for the numbers there you see f

or just mf is 2. now proceed with the

variables so x cubed x squared there is

no x for the third term

so meaning x is not gcf or just

math how about for y

for the second term is y cube third term

y but there is no y for the first term

so y is not included in the gcf

how about the z z cube for first term

z to the power of 4 for third term

so meaning z is not common to the three

terms

so only two is the gcm f

now proceed you have to find the other

factors so divide each

term by the gcmf so 28

x cubed z

cubed minus 14

x squared y cube

plus 36 y

z to the power of four divided by

[Music]

h by two

okay so

two times 28 divided by two

that is fourteen

i by the way class i divide two for the

three terms

you can also divide two by each term

that's the same answer

okay so capital x cubed

z cubed then

fourteen divided to the seven so minus 7

x squared y to the power of cube

36 divided by 2 that is

18 18 y

z to the power of four so

that is the final answer

let's have fifth example

so you have there

five h plus 40 h k cubed

okay let's check first if the two terms

are perfect cube suit

since there are two terms it could be

sum of two cubes but

check if they are perfect you so 5h that

is not a perfect cube 40h

is not also perfect cube so it might be

there is gcmf or greatest common

monomial factor among the two terms so

what would be the

gcmf

so the gcmf is 5h

okay you factor 5h for the two terms so

you divide each term 5h

divided by 5h plus

40 h

k cubed divided by 5h

so you have 5h 5h divided by 5h that is

1

plus 40 divided by 5

that is 8. h divided by 8 it's one also

eight times one is eight and then k

q okay

previously we had already

discussed the

factor of 1 plus 8 k cubed

factor for 1 plus 8 k cubed from our

example number 4

are the sum of 1 and two k

then one plus

two k plus

four k squared

that's all for sum and difference of

blue cubes stay tuned for another lesson

on factoring techniques god bless