Module 64
Special Products
3
Lesson 1
SPECIAL PRODUCTS
Hello learners! I congratulate you on having successfully mastered your
previous lessons in Math. Now that you are at this level, you will be dealing
with Intermediate Algebra. But before that, you should be equipped with some
certain understandings of Special Products. In this particular lesson, you will
develop some special rules wherein you can obtain the products in less time.
Some of these are products of binomial, square of a binomial, and the products
of the sum and difference of two terms. Moreover, this lesson will help you explore formulas
for area and investigate real-world situations such as those involving savings account. After
mastering this lesson, you should be able to find the most efficient use of time. This lesson will
also teach you to use alternative solutions in solving problems and find the right and accurate
answer in the fastest way.
Try to reflect on the following essential questions: How can you relate the methods in solving
special products and factoring to real-life problems? How will these methods help you in your
chosen career? What is the essence of learning these methods of solving special products and
factoring in your life? You should be able to arrive at the following enduring understandings:
Special products can describe the relationship of two numbers/expressions in two opposite
ways so as in real-life problems. Also, breaking down a number into its factor other than one
and itself is important and sensible in our daily lives because it will help you to develop your
logical thinking skills which might help you also in your chosen career and make you a good
citizen in the future. First things first, let me find out about your prior knowledge about this
lesson. Please take the pretest on the next page.
What do you know so far?
Are you familiar with distributive property? Do you know how to
multiply algebraic expressions? Let us found out if you are ready to
go through this lesson. Do what you are asked to do below. After
which, compare your answers with the answer key provided by your
facilitator.
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DIRECTIONS: Copy the following items and answer them correctly in your notebook.
A. Find the products of the following expressions.
B.
C.
1.
(2x – y)(3x – 5y)
4.
(r + 3p)(5r – 6p)
2.
(3m + 2)(2m – 7)
5.
(8w – 7z)(w + 2z)
3.
(5c + 4)(2c + 9)
Find the squares.
1.
(7x – 2)2
4.
(3s + 7r)2
2.
(5y + 3x)2
5.
(9v – 11w)2
3.
(b – 6c)2
(3fg + 7h)(3fg – 7h)
3
1
3
1
( x – y)( x + y)
4
2
4
2
Find the product.
1.
(6g + 7) (6g – 7)
4.
2.
(10 – 2xy)(10 + 2xy)
5.
3.
(8x + 9y)(8x – 9y)
Be Hooked!
Let us start the lesson. Take a look at the square table below and try to get its perimeter. Try
to remember the formula in getting the perimeter of a square and the distributive property you
have learned in the previous year. This will help you solve the given problem.
HINT: Perimeter of a square = 4 × side (P = 4s)
(4y
P = 4s
–3
) in
P = 4(4y – 3)
P = (16y – 12) in
As you notice, the distributive property is used when 4 is multiplied to the terms inside the
parenthesis. Therefore, the perimeter of the square table is (16y – 12) in.
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Let us have some examples and try to get the perimeter of the following figures.
(x + 9) cm
8 cm
a.
(y – 5) cm
b.
P = 2l + 2w
P = 4s
Let’s have another set of figures, but this time try to get the area of each.
(b + 2) m
c.
(c + 6) ft
d.
Area = s2
Area - lw
Take a look at my answers and compare yours. I am sure your answers somehow differ from
mine but I know you have come up with almost similar answers.
a.
P = 2l + 2w
b.
P = 2(x + 9) + 2(8)
P = 2x + 18 + 16
P = 2l + 2w
P = 4(y – 5)
P = (4y – 20) m
P = (2x + 34) cm
c.
d.
A = lw
A = s2
A = (c + 6)2
A = (b + 2))b – 2)
A = (c + 6)(c + 6)
A = b2 – 2b + 2b – 4
A = (b2 – 4) m2
A = c2 + 6c + 6c + 36
A = (c2 + 12c + 36) ft2
How was it? Did you get all the correct answers? The activity you have just done
proves that you have really mastered the lesson in simplifying algebraic expressions.
As you notice, some properties of real numbers were used. One of these is the
distributive property. Using this property makes it possible to find the factors with
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the most efficient use of time. As you go through the lesson,you will develop some special rules
wherein you can obtain the products in less time.
First, is the product of two binomials. You will learn how to multiply two binomials like
what you did in the previous exercises. You will use the distributive property and multiply
every term of the binomial by every term of the other binomial, and combine similar terms.
Moreover, one way to organize this is to use FOIL, which refers to First, Outer, Inner, and Last
terms. The term FOIL is a memory tool for applying the distributive property.
Let’s have some examples.
Example 1. Find the product of (x – 4)(x + 9).
Solution:
First
Last
(x – 4)(x + 9)
Inner
Outer
First
Outer
Inner
Last
= (x)(x) + (x)(9) – (4)(x) – (4)(9)
= x2 + 9x – 4x – 36
= x2 + 5x – 36
Thus, the product is x2 + 5x – 36.
Example 2. Find the product of (2y – 5)(y – 8).
Solution:
First
Last
(2y – 4)(y – 8)
Inner
Outer
First
Outer
Inner
Last
= (2y)(y) + (2y)(-8) – (5)(y) – (5)(-8)
= 2y2 – 16y – 5y + 40
= 2y2 – 21y + 40
Thus, the product is 2y2 – 21y + 40.
In the examples above, the product of two binomials in the form (ax + b) (cx + d) is a
quadratic trinomial. When we say quadratic trinomial, it is a second degree expression having
three terms. To obtain the first term in the quadratic trinomial you need to get the product of the
first term of the binomials. Likewise, the third term of the quadratic trinomial is the product of
the last term of the two binomials. Lastly, the middle term is the sum of the product of the inner
and the outer terms of the binomials.
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Here are some activities for you to answer. Try to apply what you have learned from the
earlier discussion.
Answer the activity below, after which compare your answers with
the answer key provided by your facilitator. You can do it. Just be
patient.
Activity A
Find the product.
1.
(3g – 4h)(g + 2h)
2.
(6x – 5)(2x – 3)
3.
(5c + 8)(3c – 1)
4.
(9y + 4z)(y + 8z)
5.
(ab – 2c)(2ab + 3c)
Another special case of products that you are going to learn is the square of a
binomial. In this case, you will multiply binomials that are alike. In short, you will
find out the squares of the given binomial.
Observe the following examples and try to look for a general pattern in getting the product.
Example 3. Find the squares.
a.
(x + 7)2 = (x + 7) (x + 7) = x2 + 14x + 49
b.
(y – 5)2 = (y – 5) (y – 5) = y2 – 10y + 25
c.
(2m + 3)2 = (2m + 3) (2m + 3) = 4m2 + 12m + 9
d.
(3k – 5)2 = (3k – 5) (3k – 5) = 9k2 – 30k + 25
e.
(4 + 3x)2 = (4 + 3x) (4 + 3x) = 16 + 24x + 9x2
What do you notice about the squares of the binomials? Do you see any pattern? Can you
explain it?
That is right! The product obtained from squaring the binomial is called a perfect square
trinomial. It is because the first term of the trinomial was obtained by getting the square of the
first term of the binomial. The middle term was obtained by multiplying the product of the first
and the last terms of the binomial by two. Lastly, the third term of the trinomial was obtained by
squaring the last term of the binomial.
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Do you have any generalization about what you have discovered?
We can now conclude that the general forms of the square of a binomial such as (a + b) and
(a – b) are,
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Try to answer the activities that follow to see if you really understand this part of the lesson.
Answer the activity below, after which compare your answers with
the answer key provided by your facilitator. You can do it. Just be
patient.
Activity B
Find the squares.
1.
(8x + 3)2
2.
(6y – 7)2
3.
(2a + 3b)2
4.
(5p – 2r)2
5.
(10 + 11m)2
Let us move on to the next one. Take a look at the following examples, and try to observe
some patterns that will help you get the product of the two binomials in less time.
Example 4. Find the product.
a.
(x – 7)(x + 7) = x2 + 7x – 7x – 49 = x2 – 49
b.
(8 + y)(8 – y) = 64 – 8y + 8y – y2 = 64 – y2
c.
(2m – 3)(2m + 3) = 4m2 + 6m – 6m – 9 = 4m2 – 9
d.
(4c + 7d)(4c – 7d) = 16c2 – 28cd + 28cd – 49d2 = 16c2 – 49d2
What did you observe? How did you get the product of the two binomials? Did you find
any pattern?
The examples above illustrate getting the product of the sum and difference of two
binomials. As you can see, to get the product of these binomials, just square the first term and
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the last term of the binomial. Eventually, the middle term will be cancelled out. So don’t worry
about the middle term. We can now say that the general form of the sum and product of two
binomials is,
(a + b) (a – b) = a2 – b2
Try to answer the activities follows. Apply the method in solving these expressions.
Answer the activity below. After which, compare your answers with
the answer key provided by your facilitator. You can do it. Just be
patient.
Activity C
Find the product of the sum and difference of the two binomials.
1.
(9k + 2y)(9k – 2y)
2.
(13s + 8r)(13s – 8r)
3.
(7b – 10c)(7b + 10c)
4.
(4m + 9)(4m – 9)
2
1
2
1
( x – y)( x + y)
5
3
5
3
5.
Let us move on to the next one. Take a look at the following examples and try to
observe some patterns that will help you get the product of a binomial and a
trinomial.
Example 4. Find the product.
a.
(x + y)(x2 – xy + y2) = x3 – x2y + xy2 + x2y – xy2 + y3 = x3 + y3
b.
(x – y)(x2 + xy + y2) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3
c.
(x – 2)(x2 + 2x + 4) = x2 + 2x2 + 4x – 2x2 – 4x – 8 = x3 – 8
d.
(3c + 1)(9c2 – 3c + 1) = 27c3 – 9c2 + 3c + 9c2 – 3c + 1 = 27c3 + 1
What did you observe? How did you get the product of a binomial and a trinomial?
Did you find any pattern?
As you can see, to get the product of this binomial and a trinomial, just use the distributive
property of multiplication wherein you will multiply the first term of the binomial by each of
the term in the given trinomial, and afterwards multiply the second term of the binomial by each
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of the terms in the same trinomial. Eventually, the middle term will be cancelled out. So don’t
worry about the middle term. We can now say that the general forms of the sum and product
of two binomials are,
(x + y)(x2 – xy + y2) = x3 + y3
(x-y)(x2 + xy + y2) = x3 – y3
To understand these further, let’s solve the following:
a.
(2a– 3b)(4a² + 6ab + 9b²)
Step 1:
Multiply the first term
of the binomial by
each term in the given
trinomial.
(2a – 3b)(4a2 + 6ab + 9b2)
= 8a3 + 12a2b + 18ab2
Step 2:
Multiply the second
term of the binomial
by each term in the
given trinomial.
= (2a–3b)(4a2 + 6ab+ 9b2)
= -12a2b– 18ab2 – 27b3
Step 3:
= 8a3 + 12a2b + 18ab2 – 12a2b – 18ab2 – 27b3
= 8a3 – 27b3
b.
Cancel out the middle
terms.
(4x + 5y)(16x2 – 20xy + 25y2)
Step 1:
Multiply the first term
of the binomial by
each term in the given
trinomial.
(4x + 5y)(16x2 – 20xy + 25y2)
= 64x3 – 80x2y + 100xy2
Step 2:
Multiply the second
term of the binomial
by each term in the
given trinomial.
(4x + 5y)(16x2 – 20xy + 25y2)
= 80x2y – 100xy2 + 125y3
Step 3:
64x3 – 80x2y + 100xy2 + 80x2y – 100xy2 + 125y3
= 64x3 + 125y3
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Cancel out the middle
terms.
Try to answer the activities that follow. Apply the method just taught in solving these
expressions.
Answer the activity below. After which, compare your answers with
the answer key provided by your facilitator. You can do it. Just be
patient.
Activity D
Find the product of the binomial and the trinomial.
1.
(a + b)(a2 – ab + b2)
2.
(2 + y)(4 – 2y + y2)
3.
(x – 3)(x2 + 3x + 9)
4.
(5ab + 3xy)(25a2b2 + 15abxy + 9x2y2)
5.
(3m – 5n)(9m2 – 15mn + 25n2)
Let us move on to the next one. Take a look at the following examples and try to
observe some patterns that will help you get the product of a binomial and a
trinomial.
Example 4. Find the product.
a.
(a + b)3 = a3 + 3a2b + 3ab2 + b3
b.
(x + 3)3 = x3 + 3(x2)(3) + 3(x)(3)2 + (3)3
= x³ + 9x² + 27x + 27
c.
(a – b)3 = a3 – 3a2b + 3ab2 – b3
d.
(2x – 3)3 = (2x)3 – 3(2x)2(3) + 3(2x)(3)2 – (3)3
= 8x³- 36x² + 54x- 27
What did you observe? How did you get the product of a binomial and a trinomial?
Did you find any pattern?
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The examples above are the product of a cube of binomial. To get the product of the cube
binomial, just use the following:
A cubed binomial (sum) is equal to the cube of the first, plus three times the square of the
first multiplied by the second, plus three times the first multiplied by the square of the second,
plus the cube of the second.
(a + b)³ = a3 + 3a2b + 3ab2 + b3
A cubed binomial (difference) is equal to the cube of the first, minus three times the square
of the first multiplied by the second, plus three times the first multiplied by the square of the
second, minus the cube of the second.
(a – b)3 = a3 – 3a2b + 3ab2 – b3
To understand these further, let us solve the following:
a.
(x + 2)3 = (x)3 + 3(x)2(2) + 3(x)(2)2 + (2)3
b.
(3x – 2)3 = (3x)3 – 3(3x)2(2) + 3(3x)(2)2 – (2)3
c.
= 27x3 – 54x2 + 36x – 8
(x + 5)3 = (x)3 + 3(x)2(5) + 3(x)(5)2 + (5)3
d.
= x3 + 6x2 + 12x + 8
= x3 + 15x2 + 75x + 125
(4x + 1)3 = (4x)3 – 3(4x)2(2) + 3(4x)(1)2 – (1)3
= 256x3 – 96x2 + 12x – 1
Try to answer the activities that follow. Apply the method just taught in solving these expressions.
Answer the activity below. After which, compare your answers with
the answer key provided by your facilitator. You can do it. Just be
patient.
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Activity E
Find the cube of each binomial.
1.
(2a + 3b)3
2.
(5y – 5)3
3.
(x – 3)3
4.
(4b + 6)3
5.
(3m – 5n)3
At this point you will have realized the importance of the methods in multiplying
binomials because they really make you find products easier and faster. Does this
lesson make any sense in your daily life? How can you relate this to your real-life
problems?
The methods in solving special products help you to make use of your time wisely. They
also teach you to use alternative solutions in solving problems. No matter how hard the problem
is or how long the process is, at the end you will succeed. This only teaches you to take all the
possible answers to your questions and find the right and accurate answers in the fastest way
possible.
What have you learned so far?
At this point, you have learned a lot from this lesson. After an in-depth discussion, let us
summarize what you have learned so far.

To find the product of two binomials, use the FOIL method which stands for First, Outer,
Inner, and Last terms. This is a memory tool for applying the distributive property. Then
simplify the outer and the inner terms.

To get the square of a binomial, just get the square of the first and the last terms and multiply
the product of the first and the last terms by two.

To solve for the product of the sum and difference of two binomials, just multiply the first
and the last terms of the binomials.

To help you in exploring formulas for area and investigate real-world situations such as
those that involve carpentry and the like, you need to learn this lesson.
Now, I want you to answer this posttest. After which, compare your
answers with mine. Give your best shot!
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Posttest
A. Find the product.
B.
C.
1.
(2m– 3n)(5m + 2n)
4.
(3b + 5c)(b + 8c)
2.
(7x–y)(2x–y)
5.
(4a– 2b)(2a + 3b)
3.
(9 + 2d)(8 –d)
Find the squares.
1.
(8 + 5k)2
4.
(7r – 2s)2
2.
(5x– 9y)2
5.
(11v + xy)2
3.
(2a + 3b)2
Find the product of the sum and difference of the two binomials.
1.
(10f + 3g)(10f– 3g)
4.
2.
(5x + 8y) (5x– 8y)
5.
3.
(6a– 14b)(6a + 14b)
(17 + cd)(17 –cd)
1
3
1
3
( a – b)( a + b)
2
4
2
4
D. Find the product of the binomial and the trinomial.
E.
1.
(2x + 3y)(4x²- 6xy + 9y²)
4.
(d–e)(d² + de + e²)
2.
(5a– 2b)(25a² + 10ab + 4b²)
5.
(6s + 5t)(36s²- 30st + 25t²)
3.
(x + 4)(x²- 4x + 16)
Find the cube of each binomial.
1.
(x + 2)3
4.
(6x– 7)3
2.
(3x- 2)3
5.
(5r + 6s)3
3.
(2x + 5)3
It is now time for you to apply what you have learned from this lesson to your life by doing
this activity. Do as directed.
PERFORMANCE TASK:
Perform the following task. Be guided by the rubric below.
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GOAL
: Your goal is to find the amount of money in the bank if the interest rate is 3% at
the end of 2 years.
ROLE
: You are a parent who wants to save money for your child’s educational plan.
AUDIENCE : The audience is your family and friends.
SITUATION : If you deposited Php5000 at the beginning of each of two consecutive years.
Suppose your bank pays interest at the rate of r, use the expression (1 + r)
(2 + r)5000 to find the amount in your account at the end of two years.
PRODUCT PERFORMANCE AND PURPOSE:
You need to present the data you collected in a short bond paper. Your document should
be type-written. Include an explanation addressed to your audience. You need to show the
expression used to find the amount in the account in standard form and write the interest rate
as a decimal.
STANDARDS AND CRITERIA FOR SUCCESS
Adequate
(2 pts)
Data is considerably
accurate with 1–2
errors.
Fair
(1 pt)
Data is minimally
accurate with 3 or
more errors.
Analysis shows
conceptual
understanding of the
problem.
Analysis shows NO
understanding of the
problem.
Explanation
A complete response
with a detailed
explanation.
Good solid
response with clear
explanation.
Explanation is
unclear.
Method of Solving
Solution completely
addresses all
mathematical
components of the
task.
Solution addresses
1 – 2 mathematical
components of the
task.
Solution does
not addresses
mathematical
components of the
task.
Criteria
Accuracy
Excellent
(3 pts)
Data is accurate with
no errors at all.
Analysis shows
Understanding of the complete
understanding of the
given task
problem.
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REFERENCES
Bautista, E.P. et al. (2006). XP Introductory Algebra I. Quezon City: Vibal Publishing House, Inc.
Bautista, E.P. et al. (2006). XP Intermediate Algebra II. Quezon City: Vibal Publishing House, Inc.
Cariño, I.D. (1999). Elementary Algebra for High School II. Pasig City: Anvil Publishing Inc.
Bellman, A. et al. (1998). Algebra. New Jersey: Prentice Hall, Inc.
Oronce, O.A. & Mendoza, M.O. (2007). E-Mathematics Elementary Algebra. Quezon City: Rex
Bookstore.
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