Factoring Part 1 - Common Monomial Factoring | Grade 8 Q1 @MathTeacherGon

Summary

In this educational video, the teacher introduces common monomial factoring, a method to break down polynomials into factors. The process involves finding the greatest common factor (GCF) of the terms in a polynomial and then dividing each term by the GCF to obtain the factors. Two examples are provided: factoring 8x^2 + 12x by finding the GCF of 4x and 12x, resulting in 4x(2x+3). The second example involves factoring a more complex polynomial, 14a^3b^2 - 8a^2b + 6a, leading to 2a(7a^2b^2 - 4ab + 3). The video concludes with a reminder to subscribe for more educational content.

Takeaways

• 📚 The video is focused on teaching common monomial factoring.
• 🔄 Factoring is described as the reverse process of multiplying polynomials.
• 📐 The process involves breaking down polynomials into factors.
• 📝 Two examples are provided to illustrate the factoring process.
• 🔢 The first example involves factoring the expression 8x^2 + 12x.
• 🔑 Finding the greatest common factor (GCF) is crucial in factoring.
• 🧮 Prime factorization is used to determine the GCF.
• 📉 The GCF for the first example is calculated to be 4x.
• ✅ Verification of the factored expression is done by multiplying the factors.
• 📑 The second example is more complex, involving the expression 14a^3b^2 - 8a^2b + 6a.
• 🔍 The GCF for the second example is found to be 2a.
• 📝 The final factored form of the second example is 2a(7a^2b^2 - 4ab + 3).

Q & A

What is the main topic discussed in the video?

The main topic discussed in the video is common monomial factoring, which is the process of breaking down a polynomial into factors by identifying the greatest common factor of its terms.

What is the reverse process of factoring?

The reverse process of factoring is multiplying polynomials, where you combine factors to form a polynomial.

What are the two examples used in the video to demonstrate common monomial factoring?

The two examples used are 8x^2 + 12x and 14a^3b^2 - 8a^2b + 6a.

How does one find the prime factors of 8?

The prime factors of 8 are found by dividing 8 by 2, which gives 4, and then dividing 4 by 2, which gives 2. So, the prime factors are 2, 2, and 2.

What is the greatest common factor (GCF) of 8x^2 and 12x?

The GCF of 8x^2 and 12x is 4x, as 4 is the highest number that divides both 8 and 12, and x is the common variable factor.

How is the GCF used to factor the expression 8x^2 + 12x?

The GCF 4x is factored out of 8x^2 + 12x, and the remaining terms are placed inside parentheses, resulting in 4x(2x + 3).

What is the GCF of the terms 14a^3b^2, 8a^2b, and 6a?

The GCF of the terms 14a^3b^2, 8a^2b, and 6a is 2a, as 2 and a are the common factors in all terms.

How does the video verify the correctness of the factored form?

The video verifies the correctness of the factored form by using the distributive property to multiply the factors back together and checking if it results in the original polynomial.

What is the final factored form of the second example, 14a^3b^2 - 8a^2b + 6a?

The final factored form of the second example is 2a(7a^2b^2 - 4ab + 3).

What is the significance of the greatest common factor in factoring polynomials?

The greatest common factor is significant in factoring polynomials because it simplifies the expression by identifying the common factors that can be factored out, making the polynomial easier to work with.

Outlines

00:00

📚 Introduction to Common Monomial Factoring

In this segment, the teacher introduces the concept of common monomial factoring, which is the reverse process of multiplying polynomials. The goal is to break down a polynomial into its factors. Two examples are provided to illustrate the process: 8x^2 + 12x and 14a^3b^2 - 8a^2b + 6a. The teacher explains the importance of finding the greatest common factor (GCF) by examining the prime factors of each term. For the first example, the GCF is determined to be 4x, and the process of dividing each term by the GCF to find the factors inside the parentheses is demonstrated. The teacher also shows how to verify the factoring by using the distributive property to check if the original polynomial can be reconstructed from the factors.

05:01

🔍 Factoring a More Complex Polynomial

This part of the video script delves into a more complex polynomial with three terms: 14a^3b^2 - 8a^2b + 6a. The teacher guides viewers through the process of finding the GCF for each term, which involves identifying the common prime factors and variables. The GCF for this polynomial is found to be 2a. The teacher then demonstrates how to divide each term by the GCF to obtain the factors inside the parentheses. The process involves simplifying the terms and canceling out common factors. The final factored form of the polynomial is 2a(7a^2b^2 - 4ab + 3). The teacher concludes by encouraging viewers to learn from the video and to subscribe to the channel for more educational content.

Keywords

💡Common Monomial Factoring

Common monomial factoring is a mathematical process where you identify and extract the greatest common factor from a set of terms within a polynomial. This process is essential in simplifying polynomial expressions. In the video, the teacher demonstrates this by factoring out the greatest common factor from the terms '8x squared' and '12x', which is '4x'. This step is crucial as it simplifies the polynomial and makes it easier to work with.

💡Factoring

Factoring, as mentioned in the video, is the reverse process of multiplying polynomials. It involves breaking down a polynomial into a product of simpler polynomials or monomials. The video's theme revolves around factoring, specifically common monomial factoring, as a method to simplify and understand polynomial expressions better. The teacher uses examples to show how factoring can be applied to different types of polynomials.

💡Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. In the video, polynomials are the main objects being manipulated. The script includes examples of polynomials like '8x squared plus 12x' and '14a cubed b squared minus 8a squared b plus 6a', which are then factored to illustrate the concept.

💡Prime Factors

Prime factors are the prime numbers that multiply together to give the original number. In the context of the video, finding prime factors is a step in determining the greatest common factor of terms within a polynomial. The teacher explains how to find the prime factors of numbers like 8 and 12 to help identify the common monomial factor.

💡Greatest Common Factor (GCF)

The greatest common factor, or GCF, is the largest factor that two or more numbers share. In the video, the GCF is used to simplify polynomials by factoring out the common elements from all terms. The teacher calculates the GCF for the terms '8x squared' and '12x' as '4x', which is then used to factor the polynomial.

💡Distributive Property

The distributive property is a fundamental algebraic property that allows for the multiplication of a term by each term within a parenthesis. In the video, the teacher uses the distributive property to verify the correctness of the factoring process by multiplying the factors back to see if they yield the original polynomial.

💡Variable

In algebra, a variable is a symbol, usually a letter, that represents an unknown value. The video discusses variables such as 'x' and 'a' in the context of polynomials. The teacher shows how to factor out common variables, like 'x' in '8x squared' and '12x', to find the GCF.

💡Exponent

An exponent is a number that indicates how many times a base number is multiplied by itself. In the video, the teacher references exponents when discussing terms like 'a cubed' and 'b squared', which represent the base 'a' multiplied by itself three times and base 'b' multiplied by itself twice, respectively.

💡Multiplication

Multiplication is one of the basic arithmetic operations, and in algebra, it's used to combine like terms or to verify factoring by using the distributive property. The video script includes multiplication as a step in both the factoring process and the verification of the factored form.

💡Division

Division is the process of splitting a number into equal parts. In the context of the video, division is used to simplify terms by separating the greatest common factor from the original polynomial. The teacher demonstrates dividing each term by the GCF to isolate the remaining factors within the parentheses.

Highlights

Introduction to common monomial factoring

Factoring is the reverse process of multiplying polynomials

First example: 8x squared plus 12x

Second example: 14a cubed b squared minus 8a squared b plus 6a

Finding the greatest common factor (GCF) of the terms

Prime factors of 8 and x squared

Prime factors of 12x

GCF is two times two (4) and x

Dividing each term by the GCF to find the factors

Verification of the factored form by multiplying

Moving on to the second polynomial

Finding the GCF of 14a cubed b squared and 8a squared b

Prime factors of 6a

Identifying common prime factors

GCF is two a

Dividing each term by the GCF to get the factors

Final factored form of the second polynomial

Encouragement to like, subscribe, and hit the notification button

Closing with a reminder of the channel's educational purpose

Transcripts

hi guys hi guys hi guys hi guys hi guys

it's me teacher

in today's video

hi guys it's me teacher doing in today's

video

we will talk about common monomial

factoring by the way

factoring is the reverse process of

multiplying polynomials in which

you're trying to break down the given

polynomial

into factors so without further ado

let's do this topic so basically guys i

have here

two examples for us to understand

how common monomial factoring works

okay so the first one is 8x squared

plus 12x and the other example is

14 a cubed b squared minus

8 a squared b plus 6a

because we need to find first the

greatest common factor

of these two terms so let's start with

8x

8x squared followed by

12 x so we need to

find the prime factors of 8 and x

squared for

the number eight you need to start with

two times two

times two because prime factors

in prime factors not in ke

divided by 2 is 4 so we have time 4 and

then 4 divided by 2 is

2 so valentine 2 times 2 times 2 which

is 8 and then as for the x squared

domain

we can factor out x squared as x

times x but we're not getting case at

because we still need to find the prime

factors of 12x

as for 12x temperature we will start

with number 12

i will start with two so i still have

six because i have two times six which

is twelve

and then we can still factor out six as

two times three so as you can see two

times 2 is 4 times

3 that will give you the answer of 12.

and then for the varnish ball since

x n times x america

min acting greatest common factor

look at the given the prime factors of

each term

as you can see at least the long pairs

the common which is

two and two

for the numbers and for the variable x

epsilon one pair

the common in variable x meaning

their gcf

is equal to two times two which is four

okay

four and then for the variable x that is

x again

foreign greatest common factor

since we haven't tried two pairs of

common

uh prime factorization at melting

variable and you know

i multiply this number to this number

which is four

and then copy your x that's why this is

our greatest column factor

after determining the greatest column

factor let's move on to this one

what we will do is we will divide

all right so i'll put another thing for

x it's all about four x we're trying to

write now the answer we have

four x and

and to get the terms inside the

parentheses you need to divide this by

4x

and also this is to be divided by 4x

8 divided by 4 that is 2

[Music]

x squared over x is x

that is the quotient of this first term

to be divided

by 4x next

plus because this one is positive 12

divided by 4 that will give the answer

of 3

and as for the x variable x divided by x

is y

that's why on another variable

and this is the factor

or this is these are the factors of the

original given

8x squared plus 12x now

sir can we check whether our answer is

correct yes

using this factors checkpoint antenna

you have four x

times two x plus three

um factoring is the reverse process of

multiplying

so we will to check

whether our answer is correct we need to

use distributed property

by multiplying each term by 4x

so 4x times 2x that is 8x squared

and then 4x times 3 that will give you

the answer of 12x as you can see they

are the same

that's why our answer is correct

so let's move on with item number two

this one is quite long

we have three terms to factor out

we have 14 a cubed b squared

minus 8a squared b plus

6a so we will start getting the greatest

the factors of

14 a cubed

b squared and

8 a squared b

we have here six a for this one and

prime factors at all we will start with

two two and times seven total

yeah two times seven for the variables

in line

a times a times a because we have a cube

three times for b b times b

okay for eight is for domain

prime factors of eight are 2 times 2

times 2 a times a

then b for 6 a

that is 2 times 3 and then the variable

a

so we need to check any common one so

they have one common

prime factor which is two and then

okay for the variable we have common

variable e

[Music]

that's why the gcf or the greatest

common factor

of these terms is to

a

it's able similar write our factors this

way called

gcf

2a and then prepare the parenthesis

all you need to do to get the other

factor is to divide this terms

by your gcf divided by 2a

divided by 2a and then divided by 2a

so 40 divided by 2 is 7

8 cubed divided by a is a squared

and then copy b squared

and then negative divided by positive is

negative

8 divided by 2 is 4

a squared divided by a is a and then

copy the variable b

plus 6 divided by 2 is 3

and then cancel out a because a over a

is

1. that's why the correct

are the factors of number two

is two a times seven a squared

b squared minus four a b plus three

so i hope now you learned something from

this video on how to use

common monomial factory in factoring

polynomials so if you're new to my

channel don't forget to like and

subscribe

and hit button for you to be updated

such

latest uploads again i am teacher going

but i am silent bye