grade 8 | unit 1 factoring

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STUDY GUIDE

GRADE 8 | UNIT 1

Factoring

Table of Contents

Introduction ......................................................................................................................................... 3

Test Your Prerequisite Skills ............................................................................................................. 4

Objectives ............................................................................................................................................ 5

Lesson 1: Factoring Polynomials with a Common Monomial Factor

- Warm Up! ................................................................................................................................. 5

- Learn about It! ........................................................................................................................ 6

- Let’s Practice! .......................................................................................................................... 7

- Check Your Understanding! ................................................................................................ 12

Lesson 2: Factoring the Difference of Two Squares

- Warm Up! ............................................................................................................................... 13

- Learn about It! ...................................................................................................................... 15

- Let’s Practice! ........................................................................................................................ 17


Lesson 3: Factoring the Sum or Difference of Two Cubes

- Warm Up! ............................................................................................................................... 23

- Learn about It! ...................................................................................................................... 24

- Let’s Practice! ........................................................................................................................ 25


Lesson 4: Factoring Perfect Square Trinomials

- Warm Up! ............................................................................................................................... 31

- Learn about It! ...................................................................................................................... 32

- Let’s Practice! ........................................................................................................................ 34



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STUDY GUIDE

Lesson 5: Factoring General Trinomials

- Warm Up! ............................................................................................................................... 40

- Learn about It! ...................................................................................................................... 41

- Let’s Practice! ........................................................................................................................ 42


Challenge Yourself! .......................................................................................................................... 49

Performance Task ............................................................................................................................ 50

Wrap-up ............................................................................................................................................. 54

Key to Let’s Practice! ......................................................................................................................... 55

References ........................................................................................................................................ 56


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STUDY GUIDE

GRADE 8 | MATHEMATICS

UNIT 1

Factoring Mathematics relies on logic and creativity, and it is studied both for its theoretical and

practical applications. For some people, not just the

professionals, the pursuit to understand the essence of

mathematics lies in the study of patterns and relationships.

For grade 8 students, the pursuit to study mathematics starts

with the understanding of the concepts of polynomials.

For some, it is not outright clear when or where the application of the study polynomials

can be readily applied. However, even the simplest tasks or scenarios can be modeled

using polynomials.

Have you ever wondered how architects maximize and design a

certain floor plan of a house or a building? …or how an engineer

computes for the right mixture of cement and gravel to be used for

the foundations of skyscrapers and other infrastructures? …or simply

how a carpenter builds a cabinet or drawer using limited supplies?

In this unit, you will learn about the different factoring techniques that

will be useful in simplifying polynomials as well as their applications in the real world.

Click Home icon to go back to

Table of Contents


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STUDY GUIDE

Before you get started, answer the following items to help you assess your prior

knowledge and practice some skills that you will need in studying the lessons in this unit.

1. List all the positive integer factors of the given number.

a. 9 b. 16 c. 51

2. Perform the indicated operation.

a.

b.

c.

d.

e.

f.

3. Expand the following expressions using special product formulas.

a.

b.

c.

d.

4. Give the formulas for the following:

a. area of a rectangle

b. area of a square

c. area of a triangle

d. area of a circle

e. volume of a cube

f. volume of a rectangular prism

Computing the area and volume of two- and three-dimensional figures

Performing basic operations on polynomials

Finding special products

Test Your Prerequisite Skills


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STUDY GUIDE

At the end of this unit, you should be able to

factor completely different types of polynomials (polynomials with common

monomial factor, difference of two squares, sum and difference of two cubes,

perfect square trinomials, and general trinomials; and

solving problems involving factors of polynomials.

Something Common

Materials Needed: fishbowl, pen and paper

Instructions:

1. This activity may be played by the whole class.

2. Your teacher has prepared a fishbowl with rolled small pieces of papers having

monomials written on them.

3. Each one of you is to pick a piece of rolled paper without looking at it. You must

also not show the content of your paper to any of your classmates just yet.

4. After each of your classmates has taken his or her pick, each one of you may

open the rolled paper and take a look at the monomial you got.

5. Each one of you must then go around the classroom and find two other persons

with monomials that have a common factor (except 1) with your monomial. The

common factor may be numerical (numeral) or literal (variable) or a combination

of both. You are free to talk while in search for your partners that would form a

triad.

Lesson 1: Factoring Polynomials with a Common

Monomial Factor

Objectives

Warm Up!


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STUDY GUIDE

Examples of monomials with common factors:

and (Both have 3 as a common numerical factor.)

and (Both have as a common literal factor.)

and (Both have as a common factor.)

6. Once a triad is formed, the three must run to the teacher and show their

monomials for verification.

7. The first three triads who get verified by the teacher get a prize.

In factoring polynomials, the first factors taken out are the greatest common monomial

factors.

For example, let us factor .

One may opt to list down the common factors of both terms of the polynomial and select

the greatest factor. Nevertheless, the following steps may be helpful and easier to

perform.

Step 1: Find the GCF of the numerical coefficients and the GCF of the variables by

prime factorization.

Numerical coefficients: Variables:

6 = 3 × 2

12 = 3 × 2 × 2

ab = a ⋅ b

bc = b ⋅ c

GCF = 3 × 2 = 6 GCF = b

Learn about It!


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STUDY GUIDE

Step 2: Get the product of the GCFs for the numerical coefficients and the variables.

This will be the greatest common monomial factor of the polynomial.

⋅

Step 3: Factor out the greatest common monomial factor and divide each term by

this to find the other factor.

The other factor is .

Therefore, .

Example 1: Completely factor the polynomial .

Solution:

Step 1: Find the GCF of the numerical coefficients and the GCF of the variables by

prime factorization.

Definition 1.1: The greatest common monomial factor is the product

of the greatest common numerical factor and a second

component made up of the common variable factors,

each with the highest power common to each term.

Let’s Practice!


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STUDY GUIDE

Numerical coefficients: Variables:

45 = 3 × 3 × 5

36 = 3 × 3 × 4

a4 = a ⋅ a ⋅ a ⋅ a

a3 = a ⋅ a ⋅ a

GCF = 3 × 3 = 9 GCF = a ⋅ a ⋅ a = a3



⋅




Therefore, .

Try It Yourself!

Completely factor the polynomial – .

Example 2: Find the greatest common monomial factor of to

factor it completely.

Solution:

Step 1: Find the GCF of the numerical coefficients and the GCF of the variables.

Besides prime factorization, you may also use continuous division.


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STUDY GUIDE

The GCF of the numerical coefficients is 6.

The GCF of the variables is .



⋅




Therefore, ).


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STUDY GUIDE

Try It Yourself!

Completely factor the polynomial .

Example 3: Given that represents a positive integer, prove that it is

a composite number.

Solution: is composite if we can factor it.

Step 1: Find the GCF of the numerical coefficients and the GCF of the variables.

Besides prime factorization, you may also use continuous division.

The GCF of 75, 15, and 45 is 15.

The GCF of , , and is .



⋅




Written in factored form, .

Since we have factored , it is a composite number.


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STUDY GUIDE

Try It Yourself!

Is the polynomial a prime or composite number?

Justify your answer.

Real-World Problems

Example 4: Mr. Santos plans to construct a circular dinner table with a square

centerpiece in the center. The radius of the table should be the same

measure as the side of the square centerpiece. Write an expression for the

remaining area of the table not covered by the centerpiece in terms of the

side of the square. Write your answer in factored form.

Solution:

Step 1: Draw a figure to illustrate the given problem.

Step 2: Note that we are looking for the area not covered by the centerpiece. Hence,

we use the formula for the area of the circle and that of the square.


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STUDY GUIDE

Since the radius should be the same measure as the side of the square, we

write

Step 3: Solving for the area not covered by the centerpiece, we have

Step 4: Rewriting the equation in factored form, we have

).

Therefore the area of the table not covered by the centerpiece is given by .

Try It Yourself!

If the table in Example 4 is in the form of a square having a circular mantle and the

radius of the mantle is equal to the side of the square , what would be the

expression for the area not covered by the mantle?

1. Factor the following expressions completely.

a.

b.

c.

d.

Check Your Understanding!


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STUDY GUIDE

2. Write an expression (in factored form) for the area of the shaded region in terms

of and .

A Tale of Two Squares

Materials Needed: colored papers, ruler, pencil, scissors, paper and pen,

cartolina, markers

Instructions:

1. Form groups of three.

2. Cut two square-shaped piece from the colored

papers. One square should be larger than the

other, and the squares must be of different colors.

Label the side of the bigger square and the side of

the smaller square .

Lesson 2: Factoring the Difference of Two Squares

Warm Up!


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STUDY GUIDE

3. Trace the smaller square onto the bigger square at the corner and cut along the

trace. With your groupmates, find an expression for the area of the remaining

figure in terms of and ?

4. Draw a horizontal line segment dividing the remaining figure into two rectangles as

shown and cut along this line.

5. Move the cut out region to the right as shown.

6. Find the expressions of the dimensions (width and length) of the rectangle in terms

of and .


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STUDY GUIDE

7. Find the expression for the area of the rectangle using the dimensions you

obtained in step 6. How does this compare to the area obtained in step 3?

8. Write your answers to the expressions required and your solutions on a cartolina

and present your work in class.

A constant is a perfect square if it is obtained by multiplying a number by itself.

Observe the following examples:

Number

(x)

Square

(x2)

Perfect

Square

1 12 = 1 1

2 22 = 4 4

3 32 = 9 9

5 52 = 25 25

10 102 = 100 100

A variable is a perfect square when its exponent is even, which is divisible by 2.

Number

(x)

Square

(x2)

Perfect

Square

a a2 a2

b2 (b2)2 = b4 b4

c3 (c3)2 = c6 c6

d5 (d5)2 = d10 d10

e10 (e10)2 = e20 e20

Learn about It!


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STUDY GUIDE

In Warm Up!, the area of the new figure in step 3, is in the form of a difference of two

perfect squares.

The difference of the squares of two numbers is equal to the product of their sum and

their difference as shown by the length and width in step 5 of the activity in Warm Up!.

That is,

Let us take the case of . What are its factors?

Check if the given involves difference of two squares:

It is a binomial.

The operation involved is subtraction.

is the square of , while is the square of .

Following the format above, the factors of are the sum and the difference of the

numbers being squared, and .

Thus, .


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STUDY GUIDE

Example 1: Factor

Solution:

Step 1: Check if there is a common monomial factor.

4x2 – 25 have no common monomial factor.

Step 2: Check if the given polynomial involves a difference of two squares.

Step 3: Identify the square root of each of the terms. The square root is the number

being multiplied by itself to obtain another number.

Step 4: Write the factors as the sum and the difference of the answers in step 3.

Let’s Practice!


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STUDY GUIDE

Try It Yourself!

Factor

Example 2: Factor .

Solution:


The GCF of the numerical coefficients is 5 and the GCF of the variables is .

Therefore, the greatest common monomial factor is . We factor this out.

Step 2: Check if any of the factors involves a difference of two squares.

Step 3: Identify the square roots of each of the terms.

1 = 12


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STUDY GUIDE

Step 4: Write the factors from steps 1 and 3.

Thus, is factored as .

Try It Yourself!

Factor completely: .

Example 3: Simplify without using a calculator.

Solution:


The terms of the given expression has no common monomial factor.

Step 2: Check if any of the factors involves a difference of two squares.

Notice that the given involves the difference of two squares since 22 and 21

are squared.

Step 3: Identify the square roots of each of the terms.


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STUDY GUIDE

Step 4: Write the factors from step 3.

Thus, is equal to .

Try It Yourself!

Simplify without using a calculator.

Real-World Problems

Example 4: The difference of the areas of two limited edition

square stamps is 128 cm2. One side of the bigger

stamp is thrice the length of one side of the

smaller. Find the lengths of the sides of the two

stamps.

Solution:

Step 1: Identify the given information.

Area of bigger square – Area of smaller square = 128 cm2.


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STUDY GUIDE

Step 2: Write the equation. Remember that the area of a square is equal to the

square of the length of its side.

Step 3: Solve for the value of s by factoring.

= 128

= 128 Square 3s.

128 Subtract s2 from 9s2.

= 0 Subtract 128 from both sides of the equation.

= 0 Factor out the greatest common monomial

factor, 8.

= 0 Divide both sides of the equation by 8.

= 0 Factor the difference of two squares.

Remember that a product can only be zero if one of the factors is zero. In this

case, either is zero or is zero.

or

The measurement of length cannot be a negative value. Thus,

The length of a side of the smaller stamp is 4 cm while the length of one side

of the larger stamp is 12 cm.


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STUDY GUIDE

Try It Yourself!

What will be the length of the sides of the two stamps in Example 4, if the

difference of their areas is 512 cm2?

1. Factor the following expressions completely:

a.

b. –

c. –

d.

2. Without raising any number to a power, evaluate the following:

a.

b.

c.

3. A photo is surrounded by a frame of uniform width. The length of the outer side of

the frame is 4 inches more than the length of a side of the photo. If the area

covered by the frame alone is 80 square inches, what is the length of a side of the

photo?



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STUDY GUIDE

Special Delivery!

Materials Needed: paper and pen, stopwatch or timer

Instructions:

1. This activity may be played by the entire class with the members of the class

divided into groups of five and a person assigned as timer and facilitator.

2. The facilitator will stand in front of the class and choose one expression to be

simplified from the given table below.

3. One representative from each group will move to the back of the classroom to

answer or give the product of the expressions flashed by the facilitator.

4. The members of each group are to take turns in answering.

5. The fastest representative to reach the facilitator with the correct answer will a

get a point.

6. The group with the highest score out of the 5 items in the table will be

considered the winner.

Questions

Lesson 3: Factoring the Sum or Difference of Two Cubes

Warm Up!


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STUDY GUIDE

Recall first what a perfect cube is.

A constant is a perfect cube if it is obtained by using a number as a factor three times.

Observe the following examples:

Number

(x)

Cube

(x3)

Perfect

Cube

1 13 = 1 1

3 33 = 9 27

5 53 = 25 125

8 83 = 512 512

10 103 = 1 000 1 000

A variable is a perfect cube when its exponent is divisible by 3.

Variable

(x)

Cube

(x3)

Perfect

Cube

a a3 a3

b2 (b2)3 = b6 b6

c3 (c3)3 = c9 c9

d5 (d5)3 = d15 d15

e10 (e10)3 = e30 e30

The expression given in Warm Up! result in either a sum or a difference of two cubes

which can be factored as

Learn about It!


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STUDY GUIDE

Let us take the case of . What are its factors?

Check if the given expression involves a sum or difference of two cubes:

It is a binomial.

The operation involved is either addition or subtraction. In this case, it is subtraction.

is the cube of , while is the cube of .

Following the format above for difference of two cubes, we have and . So,

Example 1: Find the factors of 8b3 + 1

Solution:


8b3 and 1 have no common monomial factor.

Step 2: Check if the given polynomial involves the sum or the difference of two

cubes.

Let’s Practice!


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STUDY GUIDE

8b3 + 1 is a binomial

The operation involved is addition.

8b3 is the cube of , while is the cube of .

Step 3: Identify the cube root of each of the terms.

so

, so

Step 4: Form the factors using the formula with

and .

Therefore, the factors of are and .

Try It Yourself!

Find the factors of .

Example 2: Factor

Solution:


)

Step 2: Check if any of the factors involves the sum or the difference of two cubes.


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STUDY GUIDE

is a binomial.

The operation involved is subtraction.



, so

, so


Therefore,

Try It Yourself!


Example 3: Factor completely: .

Solution:


and 1 have no common monomial factor.


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STUDY GUIDE

Step 2: Check if the given is a difference of two squares.

is the square of , and is the square of . Using ,

we have

Step 3: Check if any of the factors involves the sum or the difference of two cubes.

and are binomials

The operations involved are addition and subtraction.



, so

, so


Using the formula for factoring the sum of two cubes, we have

Therefore, in factored form, .


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STUDY GUIDE

Try It Yourself!


Real-World Problems

Example 4: The volume of a Rubik’s cube with side n is equal to n3. Another Rubik’s cube

whose side measures 8 cm is placed on top of the first cube. If the sum of

their volumes is 1 843 cm3, how long is one side of the other cube?

Solution:

Step 1: Identify the given and illustrate the given problem.

Sum of the volumes = 1843 cm3.

Step 2: Write the equation. Remember that the volume of a cube is equal to the cube

of the length of one side.


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STUDY GUIDE

Step 3: Solve for the value of by factoring.

= 1 843

= 1 843 Evaluate .

= 0 Subtract 1 843 from both sides of

the equation.

= 0 Factor the difference of two cubes.

Remember that a product can only be zero if one of the factors is zero. In this

case, either or . But is no longer

factorable and definitely would not equal to zero for any positive value of n,

so we have

or .

Therefore, the length of one side of the bigger cube is 11 cm.

Try It Yourself!

What will be the length of a side of the bigger cube in Example 4, if the smaller cube

has a side whose length 5 cm and the sum of their volumes is 341 cm3?

1. Identify if the following mathematical statement is true or false:

.



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STUDY GUIDE

2. Factor the following expressions completely.

a.

b.

c.

d.

3. A wooden block is in the shape of a cube which is hollow in the inside. The hollow

space inside is also in the shape of a cube. The outside length of a side of the block is

four cm longer than a side of the cube-shaped hollow space inside. If the total volume

occupied by the wooden part of the block is 448 cm3, find the side length of the cube

and the side length of the cube-shaped hollow space inside.

Tile Four-mation

Materials Needed: cartolina, pen, and scissors

Instructions:

1. This activity may be done in groups of three.

2. Each group will be assigned a number.

3. You and your groupmates shall cut out from cartolinas the following:

4 big squares measuring 4” × 4”

8 rectangular tiles measuring 4” × 1”

16 small squares measuring 1” × 1”

Lesson 4: Factoring Perfect Square Trinomials

Warm Up!


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STUDY GUIDE

4. Each of the groups shall form the following for each round.

1 big square tile, 2 rectangular tiles, and 1 small square.

1 big square tile, 4 rectangular tiles, and 4 small squares.



4 big square tiles, 4 rectangular tiles, and 1 small square.

4 big square tiles, 8 rectangular tiles, and 4 small squares.

5. Once your group is done forming a square with the indicated pieces, shout out

your group number.

6. The teacher shall then verify if a square was formed and the pieces are correct.

7. The group that is first to get the pieces right shall receive a point.

8. The group with the most number of points after 6 rounds wins.

In Warm Up!, you were able to create squares of varying

dimensions using the tiles. Consider the square formed using 1

big square tile, 8 rectangular tiles, and 16 small squares tiles. If

the side of the 4” × 4” square is , the shorter side of the 4” × 1”

rectangle is 1 unit, and the side of the 1” × 1” square is 1 unit, the

area covered by the square can be modeled by the polynomial:

. Can you try to find the measure of one of its sides?

To solve this kind of problem, remember that the area of a square is equal to the

square of the length of its side.

So, if the length of one side of a square is x, then the area is x2. Similarly, if the length

is , then the area is . Moreover, if the length is a – b, then the

area is .

Learn about It!


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STUDY GUIDE

Such expression for the area of a square is called a perfect square. Moreover, since the

square of a + b results in a polynomial with three terms, it is called a perfect square

trinomial.

In the case of the square formed by 1 big square tile, 8 rectangular tiles, and 16 small

square tiles, the side is of length . That means the area is .

Thus, equating the two areas, we have and found a way to factor

the perfect square trinomial .

Generally for a perfect square trinomial , by working backwards, we have

Definition 4.1: A perfect square trinomial is an algebraic

expression that can be written in the form


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STUDY GUIDE

Note that the first and last terms are squares of and respectively (which should always

be positive), and the middle term is positive or negative twice the product of and .

Hence, in the factored form, the operation involved depends on the sign of the middle

term.

Let us check if the given area (in square meters) in the problem, . Does it

follow the pattern for a perfect square trinomial?

In , observe that and 16 = (4)2. To check if it is a perfect square

trinomial, the middle term must be equal to twice the product of and .

Since (the middle term), the given expression for the area is a perfect square

trinomial. Factoring the expression, we have

.

Hence, we also find by factoring that the length of one side of the square is meters.

Example 1: Write as the square of a binomial.

Solution:


has no common monomial factor.

Step 2: Identify if the given polynomial is a perfect square trinomial.

Let’s Practice!


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STUDY GUIDE

The first and last terms are both perfect squares.

and

The middle term is negative two times the product of the square roots of

the first and last terms.

– –

Step 3: Use the square root of the first term of the trinomial as the first term of the

factor and the square root of the last term of the trinomial as the second

term of the factor. Square the binomial formed. (Note that the operation

involved in the factor depends on the sign of the middle term.)

and

Hence,

Try It Yourself!

Write 4 as the square of a binomial.


Solution:



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STUDY GUIDE

Step 2: Identify if the trinomial factor is a perfect square.

The first and last terms are both perfect squares.

and

The middle term is twice the product of the square roots of the first and

last terms.

Step 3: Use the square root of the first term of the trinomial as the first term of the

factor; use the square root of the last term of the trinomial as the second

term of the factor. Square the binomial formed.

Note that the middle term is positive (addition) so the operation to be used

in the factor is addition.

and

2

Hence, .

Try It Yourself!

Factor completely:


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STUDY GUIDE

Example 3: The square of a number is .

What is the number if ?

Solution:

Step 1: List down the given information.

Square of the number

Step 2: Find the square root of by factoring.

We first check if is a perfect square trinomial.

Notice that

may be taken as the first term and is the square of

may be taken as the last term and is the square of 2

may be taken as the middle term and is twice the product of

and 2

Therefore, is in the form of where

and . Following the format for the factors of a perfect square trinomial, we

have

Hence, the number is equivalent to .

Step 3: Substitute 9 for n.

Therefore, the number is 14.


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STUDY GUIDE

Try It Yourself!

How long is one side of a square whose area is cm2?

Real-World Problems

Example 4: Mang Berting is planting palay on a square

piece of land. The land area is given by the

expression square meters.

What is the length of the side of the land if

?

Solution:

Step 1: List down the given information.

Area of the square piece of land

Step 2: Find the square root of by factoring.

is a perfect square trinomial in the form with

and because

is a perfect square; it is the square of .

is a perfect square; it is the square of .

is equal to .

Following the formula for factoring a perfect square trinomial, we have

Hence, the side of the square is equivalent to .


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STUDY GUIDE

Step 3: Substitute the value of .

Therefore, the measure of the side of the square is 6 m.

Try It Yourself!

The area of a square coaster is given by the expression cm2. What is

the length of a side of the coaster if and ?

1. Factor completely.

a.

b.

c.

2. Complete each expression to form a perfect square trinomial.

a.

b.

c.

3. A wall clock has a shape of a square. Its area is given by the algebraic expression

cm2. Find the length of a side of the clock if .



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STUDY GUIDE

Cards on the Table

Materials Needed: flaglet for each group, standard deck of cards (only the

spade and heart suits are needed)

Instructions:

1. This activity may be played by the whole class. One of your classmates shall

assume the role of game master. Divide yourselves in groups of five.

2. The game master shall take hold of the cards and shuffle them.

3. Each card shall be assigned an integer:

A = 0 7 = 7 A = 0 7 = –7

1 = 1 8 = 8 1 = –1 8 = –8

2 = 2 9 = 9 2 = –2 9 = –9

3 = 3 10 = 10 3 = –3 10 = –10

4 = 4 J = 11 4 = –4 J = –11

5 = 5 Q = 12 5 = –5 Q = –12

6 = 6 K = 13 6 = –6 K = –13

4. The game master shall then pick two cards at random, remember them, and have

them face down on the table.

5. Using the table above, the game master shall then compute the sum and the

product of the corresponding integers of the cards and tell these to the class.

Lesson 5: Factoring General Trinomials

Warm Up!


41

STUDY GUIDE

6. The groups are then supposed to try to correctly guess the card as fast they can.

7. Once a group has the answers, they should raise their flaglet to be recognized

by the game master.

8. The group that correctly guesses the two cards (with the correct number and

suit) gets a point.

9. The group that gets the most points wins.

Suppose the product of two numbers is . What are the numbers?

To factor out a general trinomial, you can use the technique called the AC method, which

is basically the “inverse” of the FOIL method.

First, identify the coefficients and of the trinomial in the form , and then

find their product.

Then find the factors of that add up to the middle term’s coefficient, . In this case,

.

Factors of 6 Sum

1, 6 7

2, 3 5

The factors of 6 that have a sum of 5 are 2 and 3.

Next, rewrite the trinomial by expanding the middle term, , into :

Learn about It!


42

STUDY GUIDE

Regroup the polynomial into two sets of binomials. Factor each binomial using their

greatest common factor. Then we factor the common binomial factor from each set to

form the two factors.

=

Thus, written in factored form, .


Solution:

Step 1: Check if there is a common monomial factor and/or if the given trinomial is a

perfect square. If so, factor accordingly.

The terms of have no common monomial factor.

is not a perfect square trinomial.

Step 2: Find .

Given , and .

Let’s Practice!


43

STUDY GUIDE

Step 3: Find the factors of 10 whose sum is the coefficient of the middle term, 7.

Factors of 10 Sum

1, 10 11

2, 5 7

Note: When the coefficients of the middle and last terms are both positive,

the factors of must be both positive as well.

Step 4: Rewrite the middle term of the original trinomial using the factors obtained

in step 3.

Step 5: Regroup the resulting polynomial and factor each group using their greatest

common factor. Then factor out the common binomial factor from set.

=

Therefore,

Try It Yourself!

Factor .


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STUDY GUIDE


Solution:



The trinomial factor is not a perfect square.

Step 2: For the trinomial , find .

Given , and .

Step 3: Find the factors of –28 whose sum is the coefficient of the middle term, 3.

Note: When the last term is negative, one of the factors of must be

positive while the other must be negative.

Factors of –28 Sum

1, –28 –27

–1, 28 27

2, –14 –12

–2, 14 12

4, –7 –3

–4, 7 3


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STUDY GUIDE


in step 3.


common factor. Then factor out the common binomial factor from each set.

=

= –

Therefore,

Try It Yourself!

Find the factors of .

Example 3: The product of two positive integers is in the form . Without

evaluating the given expression itself, give two such integers if .

Solution: To get the numbers, we can find the factors of .



The terms of has no common monomial factor.

is not a perfect square trinomial.


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STUDY GUIDE

Step 2: Find .

Given , and .

Step 3: Find the factors of 12 whose sum is the coefficient of the middle term, –8.

Note: When the last term is positive but the middle term is negative, the

factors of must be both negative.


in step 3.



Step 6: Substitute x = 31 to find the values of the missing numbers.

Therefore, two such integers are 25 and 29.

Factors of –28 Sum

–1, –12 –13

–2, –6 –8

–3, –4 –7


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STUDY GUIDE

Try It Yourself!

The product of two positive integers is in the form . Without evaluating

the given expression itself, give two such integers if

Real-World Problems

Example 4: The floor area of a chain of boutiques is given by square

meters. Without evaluating the given expression itself, find its dimensions if

the length is , and .

Solution:



has no common monomial factor.

Also it is not a perfect square.

Step 2: To factor , find .

written in standard form is . Thus, and .

Step 3: Find the factors of 75 whose sum is the coefficient of the middle term, 28.

Factors of 75 Sum

1, 75 76

–1, –75 –76

3, 25 28


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STUDY GUIDE


in step 3.



Step 6: Evaluate the length and the width using .

Since the length is , its width would be . Finally, if ,

then the length is 27 m, while the width is 5m.

Try It Yourself!

What will be the dimensions of the floor area of the chain of boutiques in Example 4 if

the area is square meters, the length is , and ? Compute for the

required dimensions without evaluating the given expression itself.

–3, –25 –28

5, 15 20

-5, -15 -18


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STUDY GUIDE

1. Use a table to factor each of the following trinomials.

a.

b.

c.

d.

e.

2. Find all values of for which the given trinomials can be factored.

a.

b.

c.

3. Without evaluating the expression for the area, find the dimensions of a

smartphone if its area is given by , its length is and .

1. Provide an explanation for errors in factoring of the given polynomials below.

a.

b.

c. is prime (i.e. it is not factorable).

2. Is the GCF of any expression always a monomial? Give an example that supports

your answer.

3. Relate the concepts in specials products to the factoring techniques that you have

studied in this unit. Provide explanations on their relationship.

Challenge Yourself!



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STUDY GUIDE

(This activity will showcase your learning in this unit. You and two other teammates will assume the role of

packaging specialists hired by a cosmetics company.)

The company is launching their new products, and as the head of a team hired by the

company, you are tapped to provide your expertise in formulating and answering

inquiries regarding the packaging of the company’s new products. You will present the

models and explain your computations to the chief executive officer (CEO) of the company

and to the head of operations and logistics department.

Specifically, you and your team are to make different prismatic (in the form of a prism)

bundles by stacking products of different sizes.

The products are packaged in boxes with the following dimensions:

The first part of the task is purely computational. In this part, you are to answer some

inquiries about the dimensions of the bundles for specific. The solutions shall not yet use

the construction of the actual boxes as well as diagrams since they may be solved purely

by operations on polynomials and factoring. You are to answer all inquiries for all of

the following bundles:

Performance Task


51

STUDY GUIDE

Bundle 1: Form a prismatic bundle by stacking 1 blue box, 2 green boxes and 1 red box.

a) Can you compute for the total base area covered by if the height of the

bundle is cm?

b) Can you provide its length and width in terms of and ?

c) If cm and cm, provide the total base area covered as well as

the length and width.

Bundle 2: Form a prismatic bundle by stacking 2 blue boxes and 4 green boxes.

a) Can you compute for the total base area covered by if the height of the

bundle is cm?

b) Can you provide its length and width in terms of and ?

c) If cm and cm, provide the total base area covered as well as

the length and width.

Bundle 3: Form a prismatic bundle with a volume equal to .

a) What are the dimensions (length, width, height) of this prismatic bundle in

terms of and ?

b) If cm and cm, compute for the length, width, and height of the

bundle.

c) What products will fit into this bundle and what are their dimensions?

Bundle 4: Form a prismatic bundle with a volume equal to .

a) What are the dimensions (length, width, height) of this prismatic bundle in

terms of and ?

b) If cm and cm, compute for the length, width, and height of

the bundle.

c) What products will fit into this bundle and what are their dimensions?

After accomplishing the computational part of task, the second part of the task is to create

the model of a bundle. Your team is to create actual models of these boxes placed

stacked to form one of these prismatic bundles. You shall choose to model only one (1)

of the bundles stated previously. The aim of the actual model is to confirm if your


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STUDY GUIDE

computations for the dimensions asked in the inquiries agree with the boxes in your

prismatic bundle models. You are free to use any material and to create your own

patterns and designs in making the boxes.

The third part of the task is the presentation of your computations and models to the

class. You may refer to the given format below as a guide for your presentation.

I. Product Name

Project Leader

Members

Date Submitted

II. Computations and Explanations

(State each problem and present your complete and detailed solutions for each

problem. Do not forget to specify the factoring concept or formula you used in solving

the problem.)

III. Prismatic Bundle Model

(Indicate the bundle you chose to model. Present a diagram of the chosen bundle with

dimensions. Present to the class the actual stacking of the boxes to form the prismatic

bundle.)

The presentation will be evaluated according to the following: explanation of the proposal,

accuracy of computations, and appropriateness of the design and model.


53

STUDY GUIDE

Performance Task Rubric

Criteria

Below

Expectation

(0–49%)

Needs

Improvement

(50–74%)

Successful

Performance

(75–99%)

Exemplary

Performance

(99+%)

Explanation of

the Design

Proposal

Explanations and

presentation of

the layout is

difficult to

understand and

is missing several

components.

Explanations and

presentation of

the layout is a

little difficult to

understand but

includes critical

components.

Explanations and

presentation of

the layout is

clear.

Explanations and

presentation of

the layout is

detailed and clear.

Accuracy of

Computations

The

computations

done are

erroneous and

do not show wise

use of the

concepts of

factoring.

The

computations

done are

erroneous and

show little use of

the concepts of

factoring.

The

computations

done are

accurate and

shows moderate

use of the

concepts of

factoring.

The computations

done are accurate

and show an

extensive use of

the concepts of

factoring.

Appropriateness

of the Designs

The model is not

useful in

understanding

the design

proposal.

The model is

somewhat useful

in understanding

the design

proposal.

The model is well-

crafted and

useful for

understanding

the design

proposal. It

showcases the

desired layout.

The model is well-

crafted and useful

for understanding

the design

proposal. It

showcases the

desired layout and

is accurately done.


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STUDY GUIDE

Key Concept Description

Greatest Common

Monomial Factor

The greatest common monomial factor is the product of the GCF

of the numerical coefficients and the GCF of the variables.

To factor an expression using the greatest common monomial

factor, follow these steps:

Step 1: Factor out the greatest common monomial factor.

Step 2: Divide each term of the expression by the greatest

comomon monomial factor to get the other factor.

Note: Finding the greatest common monomial factor is the first

step to be done in factoring any expression.

Difference of Two

Squares

Sum and Difference

of Two Cubes

Perfect Square

Trinomial

General Trinomial

To find the factors of a trinomial in the form , follow

these steps:

Step 1: Find .

Step 2: Find factors of whose sum is .

Step 3: Rewrite the middle term using the values found in

Step 2.

Step 4: Regroup the terms and take greatest common

monomial factor in each group.

Step 5: Factor out the common binomial factor from each

group to find the factors.

Wrap-up


55

STUDY GUIDE

Lesson 1

1.

2.

3. The given polynomial is a composite number since it can be factored into

.

4. )

Lesson 2

1.

2.

3. 199

4. The length of one side of the small stamp is 8 cm while the length of one side of the

big stamp is 24 cm.

Lesson 3

1.

2. –

3.

4. The length of a side of the bigger cube is 6 cm.

Lesson 4

1.

2.

3. cm

4. The measure of the side of the square is 8 cm.

Lesson 5

1.

2.

Key to Let’s Practice!


56

STUDY GUIDE

3. The two numbers are 19 and 22.

4. If , then the length is 8 m, while the width is 7 m.

Baron, Lorraine, et. al. Math Makes Sense 8. Pearson Education Canada, 2008.

Davison, David M., et al. Pre-Algebra. Philippines: Pearson Education, Inc., 2005.

Maths is Fun. “Algebra”. Accessed January 10, 2018.

http://www.mathsisfun.com/algebra/factoring.html

McCune, Sandra Luna and Clark, William D. Easy Algebra Step-by-Step. McGraw Hill

Professional, 2011.

McGraw-Hill Education. Glencoe Math Volume 1. McGraw-Hill Professional, 2013.

References

grade 8 | unit 1 factoring

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