Mathematics 8 Quarter 1 Module 1B: Factoring Perfect Square Trinomials and General Trinomials

Mathematics Quarter 1– Module 1B Factoring Perfect Square Trinomials and General Trinomials

CO_Q1_Mathematics8_Module1B

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Development Team of the Module and Publication Information

Mathematics – Grade 8 Alternative Delivery Mode Quarter 1 – Module 1B Factoring Perfect Square Trinomials and General Trinomials First Edition, 2020

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Development Team of the Module

Writers: Lee C. Apas, Clint R. Orcejola Language Editor: Merjorie G. Dalagan Content Evaluator: Isabelita R. Lindo Layout Evaluator: Jake D. Fraga Reviewers: Rhea J. Yparraguirre, Nilo B. Montaño, Lilibeth S. Apat, Liwayway J. Lubang, Rhodora C. Luga, Lee C. Apas, Jenny O. Pendica, Vincent Butch S. Embolode, Emmanuel S. Saga Illustrator: Fritch A. Paronda Layout Artist: Clint R. Orcejola, Fritch A. Paronda Management Team: Francis Cesar B. Bringas, Isidro M. Biol, Jr., Maripaz F. Magno, Josephine Chonie M. Obseñares, Josita B. Carmen, Celsa A. Casa, Regina Euann A. Puerto, Bryan L. Arreo, Lieu Gee Keeshia C. Guillen, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga

8

Mathematics

Quarter 1 – Module 1B “Factoring Perfect Square Trinomials and General Trinomials”

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

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This module is designed and written to help you factor polynomials completely using different techniques. In all lessons, you are given the opportunity to use your prior knowledge and skills in multiplying and dividing polynomials. Activities are also given to process your knowledge and skills acquired, deepen and transfer your understanding of the different lessons. The scope of this module enables you to use it in many different learning situations. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

This module contains the following lessons:

Lesson 1: Factoring Perfect Square Trinomials Lesson 2: Factoring General Trinomials

After going through this module, you are expected to:

1. determine patterns in factoring polynomials; and
2. factor perfect square trinomials and general trinomials completely.

What I Know

[illustration: hand holding a pencil inside a circle]

Directions: Choose the letter of the correct answer. Write your answers on a separate sheet of paper.

1. What is the result when you square a binomial?
   a. binomial
   b. sum of two squares
   c. difference of two squares
   d. perfect square trinomial

For items 2 – 4. Expand (2x − 4)² and answer what is asked.

2. What is the last term?
   a. 4x²
   b. -16
   c. 16
   d. −4x²

   
   

3. What is the middle term?
   a. 8x
   b. -8x
   c. 16x
   d. -16x

   
   

4. What is the first term?
   a. 4x²
   b. −4x²
   c. 2x²
   d. −2x²

   
   

5. Which of the following is the expanded form of (x − 5)²?
   a. x² − 25
   b. x² − 5x − 25
   c. x² − 10x + 25
   d. x² − 10x + 25

   
   

6. Which of the following is a perfect square trinomial?
   a. x² − 2x + 4
   b. 4x² + 12x + 9
   c. x² − 4y + 2y²
   d. 9x² − 6xy − y²

   
   

7. All of the following are perfect square trinomials EXCEPT one. Which is it?
   a. 4x² + 12xy + 9y²
   b. 9a² − 36a + 36
   c. x²y² − 4x²y² + 4x²y²
   d. a²b² + 4a³b + 4a²

   
   

8. What is the complete factored form of 4x² + 16xy + 16y²?
   a. (2x + 4y)²
   b. (2x + 4y)(2x − 4y)
   c. (2x − 4y)²
   d. (2x − 4y)(2x − 4y)

   
   

9. What are the two numbers whose sum is 1 and whose product is -12?
   a. 3 and -4
   b. -3 and 4
   c. -3 and -4
   d. 3 and 4

   
   

10. What are the two expressions whose sum is 7x and whose product is 10x²?
    a. 2x and 5x
    b. -2x and 5x
    c. 2x and -5x
    d. -2x and – 5x

    
    

11. If one factor of x² − 5x − 24 is x + 3, what is the other factor?
    a. x – 3
    b. x – 8
    c. x + 8
    d. 2x – 8

    
    

12. What is the complete factored form of x² + 7x + 10?
    a. (x + 2)(x + 5)
    b. (x − 2)(x + 5)
    c. (x + 1)(x + 10)
    d. (x − 1)(x + 10)

    
    

13. If one factor of 3x² + 17x + 10 is 3x + 2, what is the other factor?
    a. x − 5
    b. x + 5
    c. 3x − 2
    d. 2x + 5

    
    

14. What is the complete factored form of 2x² + 5x − 3?
    a. (2x − 1)(x − 3)
    b. (2x + 1)(x + 3)
    c. (2x − 1)(x + 3)
    d. (2x + 1)(x − 3)

    
    

15. The rectangle has an area of (2x² + 8x + 8) square units. If the length is (2x + 4) units, find the width of the rectangle.
    a. (x + 2) units
    b. (x − 2) units
    c. (−x − 2) units
    d. (−x + 2) units

    
    

Lesson 1: Factoring Perfect Square Trinomials

Another factoring technique that you are going to explore is factoring perfect square trinomials. Before you will start learning this topic, recall the pattern to square a binomial as this is very important in understanding this factoring technique. Do the following activity to refresh your learning in squaring a binomial.

What’s In

[illustration: hands holding a puzzle piece inside a circle]

Patterns in squaring binomial

1. (a + b)² = a² + 2ab + b²
2. (a − b)² = a² − 2ab + b²

Activity: Remember Me!

Directions: Following the pattern in squaring a binomial, fill-in the missing term. Write your answer on your answer sheet.

1. (x + 2)² = ____ + 4x + 4
2. (2x − 3)² = 4x² − ____ + 9
3. (3x + 4)² = 9x² + 24x + ____
4. (x − y)² = ____ − 2xy + y²
5. (2x + 3y)² = 4x² + ____ + 9y²

Questions:

1. What did you do to find the first term of the product? the second term? the last term?
2. How will you determine the sign of the middle term?
3. What do you call the product of squaring a binomial?

What’s New

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Recall that squaring a binomial is a perfect square trinomial. Say, (2x + 1)² = 4x² + 4x + 1 thus, 4x² + 4x + 1 is a perfect square trinomial. The following activity will test your ability in determining perfect square trinomial.

What’s More

Activity 1: You’re the One

Directions: Supply the missing term of the factor of the given perfect trinomials below. Write your answers on a separate sheet of paper.

1. x² − 6x + 9 = (x − _____)²
2. 4x² − 4x + 1 = (_____ − 1)²
3. 9x² + 12x + 4 = (3x + _____)²
4. 4x² + 16xy + 16y² = (2x + _____)²
5. 16a² − 24ab + 9b² = (_____ − 3b)²

Activity 2: Break it Perfectly

Directions: Factor the following completely by writing each of the perfect square trinomial as the square of a binomial. Write your answers on a separate sheet of paper.

1. y² + 20y + 100
2. k² − 8k + 16
3. 16m² + 48m + 36
4. 49b² − 14b + 1
5. 3x²y − 24xy + 48y

What I Have Learned

Complete the paragraph below by filling in the blanks with correct word/s or figure/s which you can choose from the box below. Each word or figure may be used repeatedly. Write your answers on your answer sheet.

first term	a² + 2ab + b² = (a − b)²	third term	squaring a binomial
multiplied	two	factoring	perfect square
polynomials	second term	square	a² − 2ab + b² = (a − b)²
a² + 2ab + b² = (a + b)²	Simplifying	a² − 2ab − b² = (a − b)²	technique

Factoring is an important process that helps us understand more about mathematical expressions or equations. Through (1), you can rewrite your (2) in a simpler form, and when you apply the factoring (3) to mathematical expressions or equations, it can yield a lot of useful information. You have learned in this lesson about factoring perfect square trinomial. This trinomial is a result of (4). The first term of the trinomial is the square of the (5) of the binomial. The second term is the product of the (6) and (7) of the binomial which will always be (8) by (9). The third term of the trinomial is the (10) of the (11) of the binomial. If the trinomial follows the pattern in squaring a binomial, then it is a (12). To factor this, you should recognize first, whether the (13) and (14) are (15). After recognizing whether the given trinomial is a perfect square, then you can proceed to factoring following the pattern (16) or (17).

What I Can Do

Suppose you have a square room with an area of 4x² + 16x + 16.

1. Find the binomial that represents each side of your room.
2. If the measure of the side of your room is 300 inches, what is the value of x?
3. What is the area of the square?
4. Suppose you want to tile the floor of your room; how many 30 x 30 inches tiles will be used?

Lesson 2: Factoring General Trinomials

In this lesson you will learn how to factor general trinomials. There will be two types of trinomials that you are going to deal with. First is the trinomial in the form ax² + bx + c, where a = 1, and ax² + bx + c, where a ≠ 1. Before you will start exploring this lesson, let us first reactivate your basic mathematical skills.

What’s In

Activity: The Two of Us

Think of two numbers whose sum and product are given in the table below. Write your answers on your answer sheet. Item 1 is done for you.

Item	Sum	Product	The Two Numbers
1	6	8	2 and 4
2	6	-16
3	-2	-15
4	4	-12
5	6	9
6	-5	-14

Notice that in item 1, the numbers 2 and 4 in Column 4 when added will give a sum of 6 and when multiplied will give a product of 8.

Questions:

1. How did you find the two numbers in column 4 to satisfy the conditions in columns 2 (sum) and 3(product)?
2. Was it easy to find the two numbers?
3. What does it mean when the product of the numbers is negative? positive?
4. Did you recognize a pattern or technique on how to find the two numbers given its sum and product? What is it?

What’s New

Recall that FOIL method is a technique used to remember the steps in multiplying two binomials. The acronym FOIL stands for:

F – first terms O – Outer terms I – Inner terms L – last terms

Activity: Grilling with F O I L

Directions: Given below are expressions in factored form in which both factors are binomials. Follow the process in multiplying the binomials using FOIL method and answer the questions that follow. Write your answers on your answer sheet.

F O I L

1. (x + 1)(x + 2) = x² + 2x + x + 2
   = x² + 3x + 2

F O I L 2. (x – 2)(x -3) = x² − 3x − 2x + 6 = x² − 5x + 6

F O I L 3. (x + 3)(x – 4) = x² − 4x + 3x − 12 = x² − x − 12

Questions:

1. What did you observe with the numerical coefficients of the x² term?
2. What did you notice about the last terms of the trinomial? How are the last terms of each trinomial related to the last terms of the given binomials?
3. What did you notice with the numerical coefficients of the middle terms? How are the coefficients of the middle term related to the last terms of the given binomials?
4. Suppose a trinomial is given, how are you going to find its two binomials factors?

What is It

General trinomials can be classified into two (2) ways:

1. Trinomial in the form ax² + bx + c, where a = 1; and
2. Trinomial in the form of ax² + bx + c, where a ≠ 1.

In this lesson you will first learn factoring general trinomial where a = 1. The following are some examples of trinomials of the form ax² + bx + c, where a = 1.

F O I L

1. (x + 1)(x + 2) = x² + 2x + x + 2
   = x² + 3x + 2

F O I L 2. (x – 2)(x -3) = x² − 3x − 2x + 6 = x² − 5x + 6

F O I L 3. (x + 3)(x – 4) = x² − 4x + 3x − 12 = x² − x − 12

[transcriber’s note: the printed folios on these eight pages read 13 to 20; the page markers below follow the requested numbering, 17 to 24]

x² + 5x + 6 x² − 7x + 12 x² + 2x − 15

Trinomials of this form are the product of two binomials having leading coefficients of 1. Consider the illustration below where the FOIL method is being applied in multiplying two binomials having leading coefficients of 1.

[chart: FOIL diagram; no number printed]

F O I L (x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2

Sum of 2 and 1 Product of 2 and 1

Notice that the coefficient of the middle term is the sum of the last terms of the two binomials and the third term is the product of the last terms of the two binomials. If you are going to factor trinomials of the form ax² + bx + c, where a = 1, you will reverse the FOIL method.

These are the steps in factoring this trinomial.

1. Find two factors with a product equal to the last term (c) of the trinomial and a sum equal to the middle term (b) of the trinomial.
2. Write the factored form of the trinomial following the pattern: (x ± first factor)(x ± second factor)

The sign in the last term of the binomial factors can be minus signs, depending on the signs of b and c.

Let us take the following examples.

Example 1: Factor x² + 7x + 10.

Solution:

Step 1. Find two factors with a product equal to the last term (c) of the trinomial and a sum equal to the middle term (b) of the trinomial.

Here, you are going to find two factors whose product is 10 (last term) and whose sum is 7 (middle term).

Refer to the illustration below.

Product	Sum
(1)(10) = 10	1 + 10 = 11
(2)(5) = 10	2 + 5 = 7

This is the correct combination. So, 2 and 5 are the factors.

Step 2: Write the factored form of the trinomial following the pattern (x + first factor)(x + second factor).

x² + 7x + 10 = (x + 2)(x + 5)

Example 2: Factor x² + 2x − 15.

Solution:

Step 1: Find two factors with a product equal to the last term (c) of the trinomial and a sum equal to the middle term (b) of the trinomial.

Here, you are going to find two factors whose product is −15 (last term) and whose sum is 2 (middle term). Since the product is negative, the two numbers must have different signs. And since the sum is positive, the bigger number (the number with greater absolute value) must be positive.

Refer to the table below.

Product	Sum
(−1)(15) = −15	−1 + 15 = 14
(−3)(5) = −15	−3 + 5 = 2

This is the correct combination. So, −3 and 5 are the factors.

Step 2: Write the factored form of the trinomial following the pattern (x + first factor)(x + second factor).

x² + 2x − 15 = (x − 3)(x + 5)

Example 3: Factor x² − 5x − 24.

Solution:

Step 1. Find two factors with a product equal to the last term (c) of the trinomial and a sum equal to the middle term (b) of the trinomial.

Here, you are going to find two factors whose product is −24 (last term) and whose sum is −5 (middle term). Since the product is negative, the two numbers must have different signs. And since the sum is also negative, the bigger number (number with the greater absolute value) must be negative.

Refer to the table below.

Product	Sum
(1)(−24) = −24	1 + (−24) = −23
(2)(−12) = −24	2 + (−12) = −10
(3)(−8) = −24	3 + (−8) = −5

This is the correct combination. So, 3 and −8 are the two factors.

Step 2. Write the factored of the trinomial following the pattern (x + first factor)(x + second factor).

x² − 5x − 24 = (x + 3)(x − 8)

Example 4: Factor x² + 6x + 14.

Solution

Step 1. Find two factors with a product equal to the last term (c) of the trinomial and a sum equal to the middle term (b) of the trinomial.

Here, you are going to find two factors whose product is 14 (last term) and whose sum is 6 (middle term). Since the product is positive and the sum is also positive so both numbers must also be positive.

Refer to the table below.

Product	Sum
(1)(14) = 14	1 + 14 = 15
(2)(7) = 14	2 + 7 = 9

Based on the table, all the possible factors of 14 were already listed. However, there are NO two numbers having a product of 14 and the sum of 6. Thus, the given polynomial, x² + 6x + 14, is a PRIME. However, it can be factored using different method to be discussed in other lesson.

Let us discuss the second classification of general trinomial ax² + bx + c, where a ≠ 1.

Consider now factoring trinomials in which the coefficient of the squared term is other than one such as the following:

6x² − 5x − 6 3x² + 17x + 10

Trinomials of these forms also have two binomial factors in which you need to consider the ax² term (first term of the trinomial), the bx term (second term of the trinomial) and the c term (third term of the trinomial) in getting the two binomial factors.

There are many ways of factoring this type of trinomials. One of those is through trial and error. Here is an example.

Example 1: Factor 6x² − 5x − 6 through trial and error.

Solution:

Give all the factors of 6x² and −6.

Factors of 6x²	Factors of −6
(3x)(2x)	(3)(−2)
(6x)(x)	(−3)(2)
	(1)(−6)
	(−1)(6)

Write all the possible factors using the values above and determine the middle term which is −5x by getting the sum of the product of the outer terms and the product of the inner terms in FOIL method.

Possible Factors	Product of the Outer Terms	Product of the Inner terms	Sum of the product of the outer terms and the product of the inner terms
(3x − 2)(2x + 3)	(3x)(3) = 9x	(−2)(2x) = −4x	9x + (−4x) = 5x
(3x + 3)(2x − 2)	(3x)(−2) − 6x	(3)(2x) = 6x	−6x + 6x = 0
(3x − 3)(2x + 2)	(3x)(2) = 6x	(−3)(2x) = −6x	6x + (−6x) = 0
(3x + 2)(2x − 3)	(3x)(−3) = −9x	(2)(2x) = 4x	−9x + 4x = −5x
(3x + 1)(2x − 6)	(3x)(−6) = −18x	(1)(2x) = 2x	−18x + (2x) = −16x
(3x − 6)(2x + 1)	(3x)(1) = 3x	(−6)(2x) = −12x	3x + (−12x) = −9x
(6x + 3)(x − 2)	(6x)(−2) = −12x	(3)(x) = 3	−12x + 3x = −9x
(6x − 2)(x + 3)	(6x)(3) = 18x	(−2)(x) = −2x	18x + (−2x) = 16x
(6x − 3)(x + 2)	(6x)(2) = 12x	(−3)(x) = −3x	12x + (−3x) = 9x
(6x + 2)(x − 3)	(6x)(−3) = −18x	(2)(x) = 2x	−18x + 2x = −16x
(6x + 1)(x − 6)	(6x)(−6) = −36x	(1)(x) = x	9x + (−4x) = 5x
(6x − 6)(x + 1)	(6x)(1) = 6x	(−6)(x) = −6x	6x + (−6x) = 0

With the factors above, (3x + 2)(2x − 3) has the sum of the product of the outer terms and the product of the inner terms of −5x, thus making it as the factors of the trinomial 6x² − 5x − 6.

Factoring using trial and error is a long process. Knowing another way of factoring trinomials of ax² + bx + c where a ≠ 1 is very important and it is up to you which method you are going to use.

Another way of factoring this kind of trinomial is by grouping.

The following are the steps in factoring trinomials of the form ax² + bx + c where a ≠ 1.

1. Multiply the first term and the last term of the trinomial.

(ax²)(c) = (ac)x² (ac) = constant

2. Get the possible factors of the product of the first term and the last term of the trinomial in such a way that their sum is equal to the second term of the trinomial.

[chart: factor-tree diagram; no number printed]

(ac)x² nx | mx

Where nx and mx are the factors of acx². And,

nx + mx = bx (second term of the trinomial)

3. Replace the middle term (bx) by the two factors.

ax² + bx + c = ax² + nx + mx + c

4. Group ax² + nx + mx + c as follows,

(ax² + nx) + (mx + c)

5. Factor out the greatest common monomial factor of each group such that you can obtain the same binomial factor.

   
   

6. Combine the greatest common monomial factor of each group and multiply it to same binomial factor obtained in step 5. The result serves as the factors of the trinomial.

   
   

Example 1: Factor 6x² − 5x − 6

Solution:

Step 1: Multiply the first term and the last term.

(6x²)(−6) = −36x²

Step 2: Get the possible factors of the product of the first term and the last term of the trinomial in such a way that the sum will be equal to the second term of the trinomial.

Here, the product is −36x² and the sum is −5x which is the middle term. Since the product is negative, the two numbers must have different signs. And since the sum is also negative, the bigger number (number with the greater absolute value) must be negative.

Refer to the table below.

Product	Sum
(x)(−36x) = −36x²	x + (−36x) = −35x
(2x)(−18x) = −36x²	2x + (−18x) = −16x
(3x)(−12x) = −36x²	3x + (−12x) = −9x
(4x)(−9x) = −36x²	4x + (−9x) = −5x
(6x)(−6x) = −36x²	6x + (−6x) = 0

This is the correct combination, so 4x and −9x are the two factors.

Step 3: Replace the middle term in such a way that ax² + bx + c = ax² + nx + mx + c, where nx and mx are the factors. Here, the factors are 4x and −9x

6x² − 5x − 6 = 6x² + 4x − 9x − 6

Step 4: Group ax² + nx + mx + c in this pattern (ax² + nx) + (mx + c).

(6x² + 4x) − (9x + 6)

Note: Notice that the operation used in the second group was changed. This will happen if the operation between the two groups is minus (-). Always do this if you encounter this case. In this case -9x – 6 becomes 9x + 6.

Step 5: Factor out the greatest common monomial factor of each group such that you can obtain the same binomial factor.

(6x² + 4x) = 2x(3x + 2)

−(9x + 6) = −3(3x + 2)

2x and −3 are the GCF and 3x + 2 and 3x + 2 are the two same binomial factors.

Step 6: Combine the greatest common monomial factor of each group and multiply it to same binomial factor obtained in step 5. The result serves as the factors of the trinomial.

(2x − 3)(3x + 2)

So, 6x² − 5x − 6 = (2x − 3)(3x + 2)

Example 2: Factor 3x² + 17x + 10.

Solution:

Step 1: Multiply the first term and the last term.

(3x²)(10) = 30x²

Step 2: Get the possible factors of the product of the first term and the last term of the trinomial in such a way that the sum will be equal to the second term of the trinomial.

Here, the product is 30x² and the sum is 17x which is the middle term. Since the product and the sum are positive, the two factors should also be both positive.

Refer to the table below.

Product	Sum
(x)(30x) = 30x²	x + 30x = 31x
(2x)(15x) = 30x²	2x + 15x = 17x
(3x)(10x) = 30x²	3x + 10x = 13x
(5x)(6x) = 30x²	5x + 6x = 11x

This is the correct combination, so 2x and 15x are the two factors.

Step 3: Replace the middle term in such a way that ax² + bx + c = ax² + nx + mx + c, where nx and mx are the factors. Here, the factors are 2x and 15x

3x² + 17x + 10 = 3x² + 2x + 15x + 10

Step 4: Group ax² + nx + mx + c in this pattern (ax² + nx) + (mx + c).

(3x² + 2x) + (15x + 10)

Step 5: Factor out the greatest common monomial factor of each group such that you can obtain the same binomial factor.

(3x² + 2x) = x(3x + 2)

(15x + 10) = 5(3x + 2)

x and 5 are the GCF and (3x + 2) and (3x + 2) are the two same binomial factors.

Step 6: Combine the greatest common monomial factor of each group and multiply it to same binomial factor obtained in step 5. The result serves as the factors of the trinomial.

(x + 5)(3x + 2)

So, 3x² + 17x + 10 = (x + 5)(3x + 2)

Example 3: Factor 2x² + 5x − 3.

Solution:

Step 1: Multiply the first term and the last term.

(2x²)(−3) = −6x²

Step 2: Get the possible factors of the product of the first term and the last term of the trinomial in such a way that their sum is equal to the second term of the trinomial.

Here, the product is -6x² and the sum is 5x which is the middle term. Since the product is negative, the two factors must have different signs. And since the sum is positive, the factor with bigger coefficient (the number with greater absolute value) must be positive.

Refer to the table below.

Product	Sum
(-x)(6x) = – 6x²	(-x) + 6x = 5x
(-2x)(3x) = – 6x²	(-2x) + 3x = x

This is the correct combination, so -x and 6x are the two factors.

Step 3: Replace the middle term in such a way that ax² + bx + c = ax² + nx + mx + c, where nx and mx are the factors. Here, the factors are −x and 6x.

2x² + 5x − 3 = 2x² + 6x + (−x) − 3

Step 4: Group ax² + nx + mx + c in this pattern (ax² + nx) + (mx + c).

(2x² + 6x) + [(−x) − 3]

Step 5: Factor out the greatest common monomial factor of each group such that you can obtain the same binomial factor.

2x² + 6x = 2x(x + 3)

(−x) − 3 = −1(x + 3)

2x and -1 are the GCF and (x + 3) and (x + 3) are the two the same binomial factors.

Step 6: Combine the greatest common monomial factor of each group and multiply it to same binomial factor obtained in step 5. The result serves as the factors of the trinomial.

(2x − 1)(x + 3)

So, 2x² + 5x − 3 = (2x − 1)(x + 3)

Example 4: Factor 3x² − 17x + 10.

Solution:

Step 1: Multiply the first term and the last term.

(3x²)(10) = 30x²

Step 2: Get the possible factors of the product of the first term and the last term of the trinomial in such a way that their sum is equal to the second term of the trinomial.

Here, the product is 30x² and the sum is 17x which is the middle term.

Since the product is negative and the sum is negative, the two factors must have both negative signs.

Hmm

Since the product is positive and the sum is negative, the two factors must have both negative signs.

Refer to the table below.

Product	Sum
(-x)(-30x) = 30x²	(-x) + (-30x) = -31x
(-2x)(-15x) = 30x²	(-2x) + (-15x) = -17x
(-3x)(-10x) = 30x²	(-3x) + (-10x) = -13x
(-5x)(-6x) = 30x²	(-5x) + (-6x) = -11x

This is the correct combination, so -2x and -15x are the two factors.

Step 3: Replace the middle term in such a way that ax² + bx + c = ax² + nx + mx + c, where nx and mx are the factors. Here, the factors are −2x and −15x.

3x² − 17x + 10 = 3x² + (−2x) + (−15x) + 10 or = 3x² − 2x − 15x + 10

Step 4: Group ax² + nx + mx + c in this pattern (ax² + nx) + (mx + c).

3x² − 2x − 15x + 10 = (3x² − 2x) − (15x − 10)

Note: Notice that the operation used in the second group was changed. This will happen if the operation between the two groups is minus (−). Always do this if you encounter this case. In this case 15x + 10 becomes 15x – 10.

Step 5: Factor out the greatest common monomial factor of each group such that you can obtain the same binomial factor.

3x² − 2x = x(3x − 2)

−(15x − 10) = −5(3x − 2)

x and -5 are the GCF and (3x – 2) and (3x – 2) are the two the same binomial factors.

Step 6: Combine the greatest common monomial factor of each group and multiply it to same binomial factor obtained in step 5. The result serves as the factors of the trinomial.

(x − 5)(3x − 2)

So, 3x² − 17x + 10 = (x − 5)(3x − 2)

What’s More

Activity 1: Missing You!

Directions: Fill in the blank of the given equation. Write your answers on your answer sheet.

1. a² + 12a + 11 = (a + ____)(a + 11)
2. b² + 6b + 8 = (b + 2)(b + ____)
3. c² − 7c + 6 = (c − ____)(c − 1)
4. 2x² − 3x − 9 = (____x + 3)(x − 3)
5. 3x² + 5x − 2 = (____x − 1)(x + 2)

Activity 2: Break It to Me Gently

Directions: Factor each trinomial completely, if possible. If the polynomial is not factorable, write PRIME. Write your answers on your answer sheet.

1. c² − 6c − 40
2. e² + 10e + 16
3. h² − 5h − 24
4. 3x² − 5x − 12
5. 4x² + 4x − 15

What I Have Learned

This lesson discusses factoring general type of trinomials, which are divided into two forms. These are trinomials in the form of ax² + bx + c where a=1, and ax² + bx + c where a ≠ 1. Write what you have learned about each one, including the step-by-step process on how factoring is done. Write your answers on a separate sheet.

For ax2 + bx + c where a = 1	ax2 + bx + c where a ≠ 1.
Concepts Learned:	Concepts Learned:
Give a trinomial of the form ax2 + bx + c where a = 1 and factor it completely.	Give a trinomial of the form ax2 + bx + c where a ≠ 1 and factor it completely.

What I Can Do

Suppose you have a rectangular garden that has an area represented by 15x² − 4x − 4 square meters and its length is represented by 5x + 2 meters.

1. Find the binomial that represents the width of your garden.
2. Suppose the measure of the length is 12m, what is the measure of the width of the rectangle?
3. What is the area of the rectangular garden?
4. If you are going to plant your garden with okra which is 25cm apart, how many okras can you plant?

Assessment

Directions: Choose the letter of the correct answer. Write your answer on a separate sheet of paper.

1. Which of the following is a perfect square number?

   • A. 8
   • B. 24
   • C. 125
   • D. 121

2. All of the following expressions are perfect squares, EXCEPT one. What is it?

   • A. 12a²b²
   • B. x²y⁶
   • C. 16a²b⁶
   • D. 25x⁶y⁸

3. Which of the following is equal to (2x − 1)²?

   • A. 4x² + 4x + 1
   • B. 2x² + 2x + 1
   • C. 4x² − 4x + 1
   • D. 4x² − 4x − 1

4. Which of the following is a perfect square trinomial?

   • A. x² + 6x + 9
   • B. 4x² − 20x − 25
   • C. 4x² + 20xy + 9y²
   • D. 9a² − 6a²bc + a²b²c²

5. All of the following are perfect square trinomials, EXCEPT one. Which is it?

   • A. x² − 4xy + 4y²
   • B. 9a² + 24a − 16
   • C. 4x² − 12xy + 9y²
   • D. x²y² − 6x³y² + 9x⁴y²

6. If one factor of a perfect square trinomial is 2xy − 3x, what is the other factor?

   • A. 2y − 3x
   • B. 2y + 3x
   • C. 2xy − 3x
   • D. 2xy + 3x

7. Which of the following is equal to x² − 6xy + 9y²?

   • A. (x − 3y)²
   • B. (x + 3y)²
   • C. (2x − 3y)²
   • D. (2x + 3y)²

8. What is the complete factored form of 4a² − 4a²b + a²b²?

   • A. (2a + ab)²
   • B. (a − 2ab)²
   • C. (2a − ab)²
   • D. (ab − 2a)²

9. What is the complete factored form of 8x² − 24xy + 18y²?

   • A. 2(2x − 3y)(2x − 3y)
   • B. 2(2x − 3y)(2x + 3y)
   • C. 2(3x + 2y)²
   • D. 2(3x − 2y)²

10. Which of the following trinomials is factorable?

    • A. x² − 6x + 7
    • B. 2x² + 5x + 10
    • C. x² + 3x + 2
    • D. 3x² − 6x + 12

11. All of the following trinomials are factorable, EXCEPT one. What is it?

    • A. x² − 2x − 3
    • B. x² + 5x + 6
    • C. 2x² + 3x − 2
    • D. 2x² + 2x + 4

12. What is the complete factored form of x² − 4x − 96?

    • A. (x + 8)(x − 12)
    • B. (x − 8)(x + 12)
    • C. (x − 8)(x − 8)
    • D. (x + 8)(x + 12)

13. What is the complete factored form of 2x² − 6x − 8?

    • A. (x + 4)(2x − 2)
    • B. (x − 4)(2x + 2)
    • C. (4x + 2)(x − 2)
    • D. (4x − 2)(x + 2)

14. Which of the following trinomials has factors (3a − 2b) and (2a − b)?

    • A. 6a² − 7ab + 2b²
    • B. 6a² + 7ab − 2b²
    • C. 6a² − 7ab − 2b²
    • D. 6a² + 7ab + 2b²

15. A rectangular garden has an area by 6x² + x − 2 square meter. If the length is represented by 3x + 2, find a binomial that represents the width.

    • A. 2x + 1
    • B. x − 2
    • C. 2x − 1
    • D. x − 2

Additional Activities

Activity: Do More!

Solve each of the following. Write you answer on your answer sheet.

1. What does it mean when we say to completely factor a polynomial?
2. Discuss how you will factor 11x² + 19x − 6.
3. Explain why 16 − 8x + x² can be factored as either (4 − x)² or (x − 4)².
4. Student A gives (4x − 1)(x − 3) as the answer to a factoring problem. Student B gets (3 − x)(1 − 4x). Who has the correct answer?
5. Is −6x² − x + 2 factorable? If yes, how will you do it?

Answer Key

transcriber’s note: this page is printed with its answer tables rotated 180 degrees beneath an upright heading. The entries below follow the order in which the tables read once turned upright; the blocks are arranged as they appear on the page.

What I Know

1. D
2. C
3. D
4. A
5. C
6. B
7. D
8. A
9. B
10. A
11. B
12. A
13. B
14. C
15. A

Lesson 1: Factoring Perfect Square Trinomial

What’s In

1. x²
2. 12x
3. 16
4. x²
5. 12xy

What’s New

1. P
2. P
3. N
4. N
5. P

What’s More

Activity 1

1. 3
2. x²
3. 2
4. y⁴
5. a⁴

Activity 2

1. (y + 10)²
2. (k − 4)²
3. (4m + 6)²
4. (7b − 1)²
5. 3y(x − 4)²

What I Have Learned

1. Factoring
2. Polynomial
3. Technique
4. Squaring a binomial
5. First term
6. First term
7. Second term
8. Multiplied
9. Two
10. Square
11. Second term
12. Perfect square
13. First term
14. Second term
15. Perfect square
16. a² + 2ab + b² = (a + b)²
17. a² − 2ab + b² = (a − b)²

Note: Answers to Items 16 & 17 may be interchanged.

What I Can Do

1. 2x + 4
2. 148
3. 90 000 in² [?]
4. 100 pieces

Lesson 2: Factoring General Trinomial

What’s In

1. 2 and 4
2. −8 and 2 [?]
3. −5 and 3 [?]
4. −6 and 2 [?]
5. 3 and 3
6. 2 and −7 [?]

What’s In (answers may vary)

Answers may vary.

What’s More

Activity 1

1. 1
2. 4
3. 6
4. 2
5. 3

Activity 2

1. (c + 4)(c − 10)
2. (e + 2)(e + 8)
3. (h + 3)(h − 8)
4. (x − 3)(3x + 4)
5. (2x + 5)(2x − 3)

What I Have Learned

Answers may vary.

What I Can Do

1. 3x − 2
2. 4m
3. 48m²
4. 768 okras

Assessment

1. D
2. A
3. C
4. A
5. B
6. C
7. A
8. C
9. A
10. C
11. D
12. A
13. B
14. A
15. C

Additional Activities

1. A polynomial is completely factored if its factors are considered to be prime.
2. Discussion may vary.
3. They will just give the same product when expanded.
4. Both have the correct answers.
5. Yes, (−3x + 2)(x + 1) or (2 − 3x)(1 + x).

References

Book References

Alferes, M.S., Duro, M.A. (2004). MSA Intermediate Algebra. Quezon City. Quezon City MSA Academic Advancement Institute

Glorial, C.J., et.al. (2010). 21st Century Mathematics. Quezon City. Phoenix Publishing House Inc.

Oronce O.A and Mendoza M.O. (2014). e – Math Algebra. Metro Manila. Rex Books Store

Rees K.R., Sparks. F.W., and Rees C.S. (1990). College Algebra Tenth Edition. Pennsylvania Plaza. McGraw-Hill Publishing Company

Electronic References

Interactive Mathematics (2018). Special Products. Retrived from http://www.intmath.com/factoring-fractions/1-special-products.php

Math Planet (nd). Special products of polynomials. http://www.mathplanet.com/education/algebra-1/factoring-and-polynomials/special-products-of-polynomial

Paul’s Online Notes (2003). Factoring Polynomials. Retrieved from http://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx

Purple Math (nd). Factoring Polynomials. Retrieved from http://www.purplemath.com/modules/rtnladd2.htm

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