Grade 8 Math Q1 Ep2: Factoring Special Products

Summary

In this educational video, Teacher Joshua guides Grade 8 students through the process of factoring polynomials, focusing on special products like perfect squares and cubes. He explains how to factor the difference of two squares and cubes using the FOIL method and introduces mnemonics like 'SOP' for remembering the signs in factoring. The video includes practical examples, such as calculating the area of a picture frame and the volume needed for packing materials, to illustrate the concepts. The lesson concludes with a quick quiz to reinforce learning, emphasizing the importance of practice in mastering mathematical skills.

Takeaways

• 📘 The video is an educational session focused on grade 8 mathematics, specifically on factoring polynomials.
• 🔢 The concept of perfect squares and cubes is introduced, which are numbers or expressions that can be expressed as a power of two or three, respectively.
• 📐 The video explains the multiplication of polynomials using the FOIL method (First, Outer, Inner, Last).
• 🔍 Special products like the difference of two squares (a^2 - b^2) and the sum/difference of two cubes are discussed, highlighting their unique patterns.
• 📝 The process of factoring polynomials is likened to reversing the process of multiplication, starting from the product to find the factors.
• 🎨 An example using a square wooden panel helps illustrate the concept of factoring a difference of two squares in a practical context.
• 🧩 The video uses a Rubik's cube analogy to describe the complexity and variability of solving mathematical problems, emphasizing the importance of recognizing patterns.
• 🔄 The mnemonic 'SOP' (Same, Opposite, Positive) is introduced to help remember the signs in factoring the sum and difference of two cubes.
• 📦 A practical example involving a larger and smaller box is used to demonstrate the application of factoring a difference of two cubes.
• 📚 The video concludes with a recap of the special products covered and a call to practice these concepts through additional exercises.

Q & A

What is the focus of the lesson in the Deaf Ed TV video?

The focus of the lesson is on factoring polynomials, specifically special products like perfect squares, perfect cubes, the difference of two squares, and the sum and difference of two cubes.

What is a perfect square number?

A perfect square number is a number or expression that can be expressed as the power of two, such as 3 squared (3^2) or y raised to the sixth (y^6).

How many perfect squares are in the first five whole numbers?

The first five perfect squares are 1 (1^2), 4 (2^2), 9 (3^2), 16 (4^2), and 25 (5^2).

What is the difference of two squares in algebraic terms?

The difference of two squares is an algebraic expression of the form a^2 - b^2, which can be factored into (a + b)(a - b).

What is the mnemonic 'SOP' used for in factoring polynomials?

The mnemonic 'SOP' stands for 'Same Opposite Positive' and is used to remember the signs in the factoring of the sum or difference of two cubes.

How is the area of a square calculated?

The area of a square is calculated by multiplying the length of one side by itself, or side^2.

What is the volume of a cube with a side length of 5 inches?

The volume of a cube with a side length of 5 inches is 125 cubic inches, calculated as 5^3.

How do you factor the expression 3w^2 - 48?

The expression 3w^2 - 48 is factored by first taking out the greatest common factor, which is 3, resulting in 3(w^2 - 16), and then recognizing it as a difference of squares, factoring it into 3(w + 4)(w - 4).

What is the factored form of 16a^6 - 25b^2?

The factored form of 16a^6 - 25b^2 is (4a^3 + 5b)(4a^3 - 5b), recognizing it as a difference of two squares.

What is the volume of empty space in a larger box with a side length of 5 inches when a smaller box with an unknown side length x is placed inside?

The volume of empty space is the volume of the larger box minus the volume of the smaller box, which is 125 - x^3 cubic inches, and can be factored as (5 - x)(25 + 5x + x^2).

How does the video conclude about the approach to solving mathematical problems?

The video concludes that there are many ways to solve a problem in mathematics, and it emphasizes the importance of focusing on the steps, patterns, and processes observed in factoring polynomials.

Outlines

00:00

📚 Introduction to Factoring Polynomials

The script introduces an educational video by Teacher Joshua, aimed at enhancing grade 8 students' mathematics skills. It begins with a brief on factoring polynomials with the greatest common monomial factor and transitions into discussing special products in polynomial factoring. The video uses the area of a square to explain perfect squares and cubes, leading to an introduction of the difference of squares formula. It also reviews polynomial multiplication using the FOIL method and sets the stage for factoring by comparing it to reversing the multiplication process.

05:03

🖼️ Factoring Special Products: Difference of Squares

This section delves into the concept of factoring special products, specifically the difference of squares. It uses a practical example of creating a picture frame from a square wooden panel to illustrate the formula. The script explains how to factor expressions like \( 81 - p^2 \) by expressing them as the product of the sum and difference of the square roots of the terms. It also extends the discussion to include the factoring of expressions that are not immediately recognizable as differences of squares, such as \( 3w^2 - 48 \), by first extracting the greatest common factor.

10:05

🔢 Factoring Special Products: Difference and Sum of Cubes

The script moves on to discuss the factoring of the difference and sum of cubes. It provides a step-by-step guide on how to factor expressions like \( a^3 - b^3 \) and \( a^3 + b^3 \) by using specific patterns and mnemonics (SOP: Same, Opposite, Positive). A practical scenario involving packing styrofoam chips inside a larger box to secure a smaller box is used to demonstrate the application of the difference of cubes formula. The section reinforces the idea that factoring involves recognizing and applying specific algebraic patterns.

15:06

📦 Practical Application and Recap of Factoring Techniques

This part of the script applies the learned factoring techniques to a real-world scenario of calculating the volume of styrofoam chips needed to fill the space between two boxes. It demonstrates the factoring of \( 125 - x^3 \) as an example of a difference of cubes. The video then provides a recap of the special products discussed, including the difference of squares and the sum and difference of cubes, emphasizing the importance of recognizing patterns for effective factoring. It concludes with a set of practice problems for the students to apply their newly acquired knowledge.

20:06

🏁 Conclusion and Encouragement for Mathematical Practice

In the concluding part, Teacher Joshua summarizes the lesson, emphasizing the importance of practice in mastering the factoring of special products in algebra. He encourages students to continue practicing with the self-learning modules and to approach mathematics as a way to develop critical, logical, and responsible thinking. The script ends with a motivational note, reminding students of the next topic to be covered—factoring trinomials—and the overarching message that mathematics is an exercise for the mind.

Keywords

💡Factoring

Factoring is the process of breaking down a polynomial into a product of its factors. In the video, factoring is the main theme, focusing on special products like the difference of squares and the sum or difference of cubes. The script uses examples such as 'a squared minus b squared' to illustrate how to factor polynomials by identifying and applying patterns.

💡Perfect Square

A perfect square is a number or expression that can be expressed as the square of an integer or another expression. In the context of the video, perfect squares are used to identify patterns in polynomials that can be factored as a difference of squares, such as '3 squared' which equals 9, or 'y raised to six' which is the square of y cubed.

💡Perfect Cube

A perfect cube is a number or expression that can be expressed as the cube of an integer or another expression. The video script introduces perfect cubes to explain how to factor expressions that are cubes minus cubes or cubes plus cubes, using '8, which is equal to 2 raised to the third power' as an example.

💡Difference of Two Squares

The difference of two squares is a specific algebraic identity where the difference between the squares of two terms can be factored into the product of the sum and difference of the terms. The video uses 'a squared minus b squared' as a key example, explaining that it factors into '(a + b)(a - b)'.

💡Sum and Difference of Two Terms

This concept refers to the algebraic method of multiplying the sum and difference of two terms, which is essential for factoring polynomials like the difference of squares. The script demonstrates this with the FOIL method, where 'a plus b times a minus b' results in 'a squared minus b squared'.

💡Greatest Common Monomial Factor (GCMF)

The GCMF is the largest monomial that divides all terms of a polynomial. In the video, GCMF is used as a strategy for factoring polynomials, where a common factor is first extracted before further factoring the remaining polynomial, as shown in the example '3w squared minus 48'.

💡Special Products

Special products are specific algebraic patterns that can be factored or expanded using certain identities. The video focuses on special products like the difference of squares and the sum or difference of cubes, which are key to factoring certain types of polynomials.

💡Mnemonic

A mnemonic is a memory aid or formula used to help remember certain information. In the video, the mnemonic 'SOP' (Same, Opposite, Positive) is introduced to help remember the signs in the factoring of the sum or difference of two cubes, aiding in the memorization and application of factoring techniques.

💡Volume

Volume is the measure of the space occupied by an object. In the video, volume is used in the context of a cube to illustrate the concept of the difference of cubes when factoring. The example of 'the volume of the bigger box minus the volume of the smaller box' is used to explain how to find the volume of empty space to be filled.

💡Factored Form

Factored form refers to expressing an algebraic expression as a product of its factors. The video script repeatedly uses factored form to demonstrate the outcomes of factoring polynomials, such as '81 minus p squared' being expressed as '(9 + p)(9 - p)'.

💡Self-Learning Module

A self-learning module is a set of instructional materials designed for individual study. The video script encourages viewers to practice what they've learned through self-learning modules, which likely contain exercises and assessments to reinforce the concepts taught in the video.

Highlights

Introduction to factoring polynomials with the greatest common monomial factor

Explaining perfect squares and their mathematical representation

Demonstration of calculating the area of a square as a perfect square

Identification of perfect cube numbers and their properties

Listing the first five perfect squares and cubes of whole numbers

Recap of polynomial multiplication using the FOIL method

Introduction to the difference of two squares as a special product

Explanation of the pattern in the difference of two squares and its factored form

Illustration of factoring as the reverse process of multiplying polynomials

Practical example of calculating the area of a picture frame using factoring

Factoring a difference of two squares with a variable and constant

Introduction to the difference and sum of two cubes as special products

Demonstration of deriving the difference of two cubes through multiplication

Explanation of the mnemonic 'SOP' for remembering signs in factoring cubes

Practical application of factoring the difference of two cubes to fill a box with styrofoam

Guidance on factoring polynomials that are not perfect squares or cubes

Factoring a polynomial with a common factor and difference of squares

Introduction to the next lesson on factoring trinomials

Emphasis on the importance of practice and determination in mastering mathematics

Transcripts

[Music]

[Music]

good day everyone

and welcome to deaf ed tv i am your

teacher joshua

i'll be your guide in sharpening your

skills and enhancing your minds in order

to face

the challenges here in grade 8

mathematics

your self-learning mojo your pens and

your paper with you

and let us have a wonderful day of

learning

last time we talked about factoring

polynomials with the greatest

common monomial factor

today we will factor polynomials that we

consider

special products but before that

i would like you to observe suppose

that this square has a side that

measures 3

units what do you think is its area

the area of the square is just equal to

the product

of its side multiplied by itself

so what is 3 squared or three

times three the area of the square

is nine square units

this illustration shows an example of a

perfect

square number perfect squares are

numbers

or expressions that can be expressed to

the power

of two another example is y

raised to six since y raised to six

is equal to the square of y cubed

by the same logic perfect cube numbers

or expressions

can be expressed to the power of 3.

example is 8

which is equal to 2 raised to the third

power

now let us try to determine the first

five

perfect squares and cube whole numbers

before proceeding with factoring let us

recall how to multiply polynomials

what is the product of the sum and

difference of two terms

the quantity a plus b times the quantity

a minus b remember

that we can use the foil method to

multiply

our polynomials we distribute each term

to the other group the first term will

be the product

of the first terms of each binomial

a times a which is

a squared next is we multiply the outer

terms

positive a times negative b

is negative a b then the inner terms

positive b times positive a

is positive a b the last term is the

product

of positive b times negative b

which is negative b squared

we can simplify the polynomial by

combining like terms

which will give us a negative a b

plus a b equal to zero

hence we are left with a squared

minus b squared this is a special

product that we call

the difference of two squares

this polynomial is a special product

because there is a pattern that is so

unique

to its form multiplying polynomials is

like

driving a car forward you are given a

starting point

and you need to find the product at the

last destination

while factoring is the reverse process

of multiplying polynomials

we will drive backwards as we are given

with the product and we are asked to

find its factors

with the previous example we can say

that the factors of a difference of two

squares

a squared minus b squared is the product

of the sum and the difference of the

positive square roots of each term

the quantity of a plus b times the

quantity

of a minus b look at the square

wooden panel the length of each side

measures nine inches what is the area

of the wooden panel the area is 81

square

inches now i want to make a picture

frame

using this square panel by cutting a

square hole inside

whose shape and size is equal to the

photograph

to be placed in since i'm not yet

decided on the size of the photograph

let us just say its side measures p

inches what will be the area of the

picture frame

in factored form we know that a square

hole is to be caught in the square

wooden panel

so we will subtract the area of the

photograph

from the area of the wooden panel it is

stated that the side of the panel

measures 9 inches

so the area of that square panel is 81

square inches

then we will subtract the area of the

photo

from the area of the wooden panel the

area is

p squared what kind of polynomial is

this

this is a difference of two squares

the polynomial can be expressed as the

square of

nine minus the square

of p hence the factored form of the area

81 minus p squared is equal to the

quantity

of 9 plus p times the quantity

9 minus p but do not forget the unit

square inches how about this one

right 16 a raised to 6

minus 25 b squared in factored form

we start with checking if each term is a

perfect square

16 a raised to 6

is equal to the square of a cubed

while 25 b squared

is equal to the square of 5b

the polynomial is a difference of two

squares

since both terms are perfect squares

and the operation is subtraction hence

16 a raised to 6 minus 25 b squared

is equal to the quantity of 4a cubed

plus 5b times the quantity of 4a cubed

minus 5b there are some cases of

polynomials

that do not seem to be a difference of

two squares

since the terms are not perfect squares

but remember there are other ways to

factor

your polynomial consider this example

factor 3 w squared

minus 48 completely are the terms of

three w squared minus 48 perfect squares

no they are not what else can we do to

factor the polynomial

note that both terms have a common

factor

of 3 hence the binomial can be factored

using a combination the greatest common

monomial factor

and the sum and difference of two terms

since we have identified that the

greatest common factor

of 3w squared minus 48 is 3

we can divide the polynomial by the

common factor

so we can express it as 3

multiplied by the quantity

w squared minus 16

then observe that the polynomial factor

is a difference of two squares we can

further

show that the expression three times

the quantity of the square of w

minus the square of four

can you give me the final factored form

we can write the polynomial

three times the quantity of w

plus four times the quantity

of w minus four

do you know what this is this

is a rubik's cube invented in 1974

by erno rubik there are many variations

of this puzzle

and there are also infinite number of

ways to solve this

the world record in the fastest solve of

a rubik's cube

is 4.22 seconds by felix

zemdega just like this rubik's cube

with different sides or faces the

difference of two squares

is just one of the several special

products

the next two involve cubes the

difference of two cubes

a cubed minus b cubed

and the sum of two cubes a cubed

plus b cubed but first

let us show how to get this special

product

multiply the quantity of a minus b

times the quantity of a squared plus

a b plus b squared

we can first multiply a to each term of

the trinomial

a times a squared is a cubed

a times a b is positive

a squared b and a

times b squared is positive a

b squared next is to multiply

negative b to each term of the trinomial

do not forget the signs in multiplying

we will have negative b

times a squared is negative

a squared b then negative

b times a b is negative

a b squared and negative b

times positive b squared is negative

b cubed combining like terms

we will remove zero pairs positive

and negative a squared b and a

b squared thus we are left with

a cubed minus b cubed

we showed that the binomial a minus b

multiplied with a trinomial a squared

plus

a b plus b squared is equal to the

difference

of two cubes since they are equal

we can also say that the reverse is the

same

meaning that the factored form of the

difference of two cubes

is given by this pattern the pattern is

also similar

in factoring the sum of two cubes a

cubed

plus b cubed what do you notice

where do they differ the sign or

operation

in the binomial factor and the first

operation in the trinomial factors are

different

they both depend on the operation used

in the polynomial there is a special

trick for us to remember the signs

used in this sum or difference of two

cubes

you can use the mnemonic s o

p meaning same opposite

positive the binomial factor has the

same

sign or operation with the given the

second sign

or the first operation in the trinomial

factor

is opposite of the binomial

the last sign is always positive

regardless of the given look at these

boxes

they are both cubes i would like to put

the smaller box

inside the bigger box and to keep it

safe

i will insert some styrofoam chips that

will fill the empty space

in the bigger box how much styrofoam

chips

is needed to fill the space and

completely secure

the smaller box the bigger box has a

width of 5

inches but i do not know the size of the

smaller box

so let us represent it with x

since we are asked about the amount of

space to be filled

we are talking about the volume what

is the formula for the volume of a cube

the volume of the cube is given by the

equation v

for volume equal to s cube

where s is the length of its side or

edge so what are the volumes of the

bigger box

and the smaller box the volume of the

bigger box is equal to the cube

of 5 which is 125

cubic inches well the volume of the

smaller box

is x cubed since we didn't know the

exact measure

of the side now let us place the smaller

box

inside the bigger boss what happened to

the volume

inside the bigger box it got smaller

so the amount of space to be filled by

styrofoam chips

is just the volume of the bigger box

minus the volume of the smaller box

substituting the values that we have

the volume of the empty space is equal

to 125

minus x cubed cubic inches

we can also write this in factored form

since it is a difference of two cubes

again we can express 125

as the cube of five minus

x cubed we write the binomial factor

using the expressions we acquired

5 and x

remember that the first sign is the same

with the given

so we will have 5 minus

x then the first term of the trinomial

factor

is the square of 5 which is 25

the next operation is opposite the

binomial

what is opposite of minus it is plus

then the middle term is the product of 5

and x 5 times x is just 5x

the last sign is always positive

or a plus sign so finally the last term

is the square of x and it is

x squared the factored form of the

amount of space

the styrofoam chips need to fill is the

quantity

5 minus x times the quantity

25 plus 5 x

plus x squared how about this example

factor 40k cubed plus 5. if you notice

the operation involved is addition but

the terms are not perfect cubed

expressions

so what else can we do observe

that both terms are divisible by five

hence a greatest common monomial factor

exists

we can write the polynomial as five

times the quantity eight k cubed

plus one then we proceed to factoring

the sum of two cubes eight

k cubed is the cube of the expression

two k well one is equal to the cube of

one

next what is the binomial factor

it is two k plus one

since this is a sum of two cubes

then the first term of the trinomial is

the square

of 2k what is it

it is 4k squared the next operation

must be minus because we need the

opposite

of the previous one after that we

multiply

the terms of the binomial 2k

times 1 is 2k

the final sign is always a positive sign

so lastly the third term of the

trinomial

is one squared and the answer to that

is one so we can say that forty k

cubed plus one is equal to five

times the quantity two k plus one

times the quantity of four k squared

minus two k plus one

and there you have it that's our lesson

for today

now let us have a recap some polynomials

are called special products

because they have a certain pattern that

we can use

in factoring them the difference of two

squares

a squared minus b squared is equal to

the product of the sum

and difference of the roots of each term

a plus b and a minus b

the sum and the difference of two cubes

are the product of a binomial

and the trinomial factor whose signs

depend

in the given polynomial remember the

mnemonic

sop same

positive when writing the signs or

operations

in factoring this special products

and now that 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 5 seconds to answer

each problem

[Music]

choose the letter that contains the

correct factors

of the given polynomial number one

factor d squared minus 25

is it a the quantity d plus 5

times the quantity d minus 5

letter b the quantity d plus 25

times the quantity d minus 25

or is it c d plus five

times the quantity d squared minus

five d plus 25

or d the quantity d minus five

times the quantity d squared plus 5 d

plus 25

[Music]

the correct answer is a the quantity d

plus 5 times the quantity d minus

5 next number factor

25 e squared minus 16

is it a e plus 4

times e minus 4

b the product of 5

e plus 4 and 5 e minus 4

c the quantity of the sum 5 e plus 4

times the trinomial 25 e squared

minus 20 e plus 16

or d 5 e minus 4

times the trinomial 25 e squared plus

20d

plus 16.

[Music]

the correct answer is b five e plus four

multiplied by the quantity five e minus

four

number three factor twenty c

squared minus forty five d squared

the choices are a the quantity 2c

plus 3d times the quantity 2c

minus 3d b

the quantity 4c plus 90 times the

quantity

4c minus 9d letter c

5 times the quantity of 2c

plus 3d times 2c minus 3d

or d 5 times the quantity

of 4c plus 9d times the quantity

4c minus 9d

[Music]

correct answer is letter c 5

times the product of the binomials 2c

plus 3d

and 2c minus 3d how was the activity

did you find it difficult just keep on

practicing on the examples

and assessment found on your

self-learning modules

remember that in mathematics practice

makes you

better as an additional exercise and

practice at home

try answering the activity on your

self-learning module

about factoring special products the

difference of two squares

and the sum or difference of two cubes

i hope that you have learned a lot in

our episode today

note that there are many ways to solve a

problem and you must focus on the steps

and patterns

or the process that you have observed in

factoring these polynomials

with practice and determination i

believe that you can ace

any lesson in mathematics for our next

episode

we will be factoring polynomials with

three terms

or what we call trinomials

[Music]

remember math is not only about numbers

and operations

it is an exercise of 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 again next

time

[Music]

[Music]

you