BASILAN NATIONAL HIGH SCHOOL

Isabela City
Mathematics Department
S.Y. 2020-2021

MODULE FOR SCIENCE, TECHNOLOGY AND ENGINEERING (STE) PROGRAM

ELECTIVE MATHEMATICS 9: ADVANCED ALGEBRA
Subject

Quarter: I
Week: 6
Topic: Polynomials

I – Objectives:
 Factor polynomials ;
 Illustrate special products of polynomials.

 Determine which factors are common to all terms in an expression.

 Factor common factors.


II – Lesson Proper:
A. Content
 Factoring
The process of factoring is essential to the simplification of many algebraic expressions
and is a useful tool in solving higher degree equations. In fact, the process of factoring is so
important that very little of algebra beyond this point can be accomplished without
understanding it.
The distinction between terms and factors has been stressed. Terms are added or
subtracted and factors are multiplied. Three important definitions follow.
Terms occur in an indicated sum or difference. Factors occur in an indicated product.
An expression is in factored form only if the entire expression is an indicated product.
In Factored Form Not in Factored Form
Example 1 2𝑥(𝑥 + 𝑦) 2𝑥 2 + 2𝑥𝑦
Example 2 (𝑥 + 𝑦)(3𝑥 − 2𝑦) 3𝑥 2 − 𝑥𝑦 − 2𝑦 2
Example 3 (𝑥 + 4)(𝑥 2 + 3𝑥 − 1) 𝑥 3 + 7𝑥 2 + 11𝑥 − 4

Note in these examples that we must always regard the entire expression. Factors can
be made up of terms and terms can contain factors, but factored form must conform to the
definition above.
Factoring is a process of changing an expression from a sum or difference of terms to
a product of factors.
Note that in this definition it is implied that the value of the expression is not changed -
only its form.

 Removing Common Monomial Factors

If every term of a polynomial contains a common monomial factor, then by the
distributive law, the polynomial can be written as the product of the common monomial factor
and the quotient obtained by dividing the original polynomial by the common factor. For
instance, 𝑎 is a common monomial factor of each term of the trinomial 𝑎𝑥 + 𝑎𝑦 + 𝑎𝑧; thus
(𝑎𝑥 + 𝑎𝑦 + 𝑎𝑧) = 𝑎(𝑥 + 𝑦 + 𝑧)

To factor an expression by removing common factors proceed as in example 1.

Example 1
Factor 3𝑥 2 + 6𝑥𝑦 + 9𝑥𝑦 2 .
Solution: First list the factors of each term
3𝑥 2 has factors 1, 3, 𝑥, 𝑥 2 , 3𝑥 , 𝑎𝑛𝑑 3𝑥 2
6𝑥𝑦 has factors 1, 2, 3, 6, 𝑥, 2𝑥, 3𝑥 , 6𝑥, 𝑦, 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛.
9𝑥𝑦 2 has factors 1, 3, 9, 𝑥, 3𝑥 , 9𝑥, 𝑥𝑦, 𝑥𝑦 2 , 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛.

3𝑥 is the greatest common factor of all three terms.

Next look for factors that are common to all terms, and search out the greatest of these.
This is the greatest common factor. In this case, the greatest common factor is 3𝑥.
Proceed by placing 3𝑥 before a set of parentheses.
3𝑥 ( )
The terms within the parentheses are found by dividing each term of the original
expression by 3𝑥.
3𝑥 2 + 6𝑥𝑦 + 9𝑥𝑦 2 = 3𝑥(𝑥 + 2𝑦 + 3𝑦 2)

Note that this is the distributive property. It is the reverse of the process that we have been
using until now.

The original expression is now changed to factored form. To check the factoring keep in
mind that factoring changes the form but not the value of an expression. If the answer is
correct, it must be true that 3𝑥 (𝑥 + 2𝑦 + 3𝑦 2 ) = 3𝑥 2 + 6𝑥𝑦 + 9𝑥𝑦 2 . Multiply to see that this is
true. A second check is also necessary for factoring - we must be sure that the expression has
been completely factored. In other words, "Did we remove all common factors? Can we factor
further?"
If we had only removed the factor "3" from 3𝑥 2 + 6𝑥𝑦 + 9𝑥𝑦 2 the answer would be
3(𝑥 + 2𝑥𝑦 + 3𝑥𝑦 2 ).
2

Multiplying to check, we find the answer is actually equal to the original expression.
However, the factor x is still present in all terms. Hence, the expression is not completely
factored.
This expression is factored but not completely factored.

For factoring to be correct, the solution must meet two criteria:

1. It must be possible to multiply the factored expression and get the original
expression.
2. The expression must be completely factored.

Example 2: Factor 12𝑥 3 + 6𝑥 2 + 18𝑥.

Solution:
At this point it should not be necessary to list the factors of each term. You should be
able to mentally determine the greatest common factor. A good procedure to follow is to think
of the elements individually. In other words, don’t attempt to obtain all common factors at once
but get first the number, then each letter involved. For instance, 6 is a factor of 12, 6, and 18,
and 𝑥 is a factor of each term. Hence, 12𝑥 3 + 6𝑥 2 + 18𝑥 = 6𝑥(2𝑥 2 + 𝑥 + 3). Multiplying,
 we get the original and can see that the terms within the parentheses have no other common
factor, so we know the solution is correct.

Say to yourself, "What is the largest common factor of 12, 6, and 18?"

Then, "What is the largest common factor of x3, x2, and x?"

Remember, this is a check to make sure we have factored correctly.

Example 3 Factor 𝑎2 𝑏 2 𝑐 + 2𝑎𝑏2 𝑐 2 − 3𝑎𝑏3 𝑐.

Solution: We note that 𝑎, 𝑏2 , and 𝑐 are common factors.
Hence 𝑎2 𝑏2 𝑐 + 2𝑎𝑏2 𝑐 2 − 3𝑎𝑏3 𝑐 = 𝑎𝑏2 𝑐(𝑎 + 2𝑐 − 3𝑏).
Check: 𝑎𝑏2 𝑐 (𝑎 + 2𝑐 − 3𝑏) = 𝑎2 𝑏2 𝑐 + 2𝑎𝑏2 𝑐 2 − 3𝑎𝑏3 𝑐

Example 4 Factor 2𝑥 − 4𝑦 + 2.
Solution: The only common factor is 2.
2𝑥 − 4𝑦 + 2 = 2(𝑥 − 2𝑦 + 1)
Example 5 Factor 5𝑥 𝑦 + 10𝑥 𝑦 + 5𝑥𝑦 2
3 2 2

Solution: 5𝑥 3 𝑦 + 10𝑥 2 𝑦 2 + 5𝑥𝑦 2 = 5𝑥𝑦(𝑥 2 + 2𝑥𝑦 + 𝑦)

Example 5 Factor 2𝑥 − 4𝑦 + 2.
Solution: 5𝑥 3 𝑦 + 10𝑥 2 𝑦 2 + 5𝑥𝑦 2 = 5𝑥𝑦(𝑥 2 + 2𝑥𝑦 + 𝑦)

Again, find the greatest common factor of the numbers and each letter separately.
If an expression cannot be factored it is said to be prime.

Remember that 1 is always a factor of any expression.

 Difference of Two Squares

From the special product in the previous topic, we have the formula
𝑥 2 − 𝑦 2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
The left side of this formula is the difference of two squares, and this formula states that it can be
written as the product of the sum and the difference of two squares.
Example:
The binomial 𝑥 2 − 4 is the difference of two squares 𝑥 2 and 22 . Therefore
𝑥 2 − 4 = (𝑥 + 2)(𝑥 − 2)
When the sum of two numbers is zero, one of the numbers is said to be the additive inverse of the
other.
For example: ( + 3) + (-3) = 0, so + 3 is the additive inverse of - 3, also -3 is the additive inverse of +3.

In each example the middle term is zero. Note that if two binomials multiply to give a binomial (middle
term missing), they must be in the form of (a - b) (a + b).

The rule may be written as 𝑎2 − 𝑏2 = (𝑎 − 𝑏)(𝑎 + 𝑏). This is the form you will find most helpful in
factoring.
Reading this rule from right to left tells us that if we have a problem to factor and if it is in the form
of 𝑎2 − 𝑏2 , the factors will be (a - b)(a + b).

Solution
Here both terms are perfect squares and they are separated by a negative sign.

Where 𝑎 = 5𝑥 and 𝑏 = 4.
Special cases do make factoring easier, but be certain to recognize that a special case is just that-
very special. In this case both terms must be perfect squares and the sign must be negative, hence
"the difference of two perfect squares."
The sum of two squares is not factorable.

You must also be careful to recognize perfect squares. Remember that perfect square numbers are
numbers that have square roots that are integers. Also, perfect square exponents are even.
Students often overlook the fact that (1) is a
perfect square. Thus, an expression such as
x2 - 1 is the difference of two perfect squares
and can be factored by this method.

B. Activity:

“Why Is a Stick of Gum Like a Sneeze?”

For each exercise, multiply the two polynomials. Find your answer in the set of answers
under the exercise. Cross out the letter above your answer. When you finish, the answer to
the title question will remain!
 1)

6
 1)

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 RUBRICS
CRITERIA Outstanding Satisfactory Developing Beginning
5 4 3 2
Correctness The message 1-2 words are More than 2 The message
of the is clear and missing. words are cannot be
message complete. missing. understood. 1)
Accuracy All solutions All solutions Some of the No solution is
are correct are correct. solutions are presented.
and shown in correct.
detail.
Neatness and The work is The work is The work is The work
Organization presented in a presented in a presented in appears
neat, clear neat, clear a neat, clear sloppy and
and organized and organized and unorganized.
fashion that is fashion that is organized It is hard to
easy to read. usually easy fashion but know what
to read. may be hard information
to read at goes together.
times.
C. Synthesis
Remember!!!
 Distributive laws can be used to write the product of a monomial and a
polynomial.
 Steps in Multiplying Polynomials:
Step 1: Distribute each term of the first polynomial to every term of the
second polynomial. Remember that when you multiply two terms together you
must multiply the coefficient (numbers) and add the exponents.
Step 2: Combine like terms (if you can).
 FOIL Method stands for First, Outer, Inner, Last.
 Special Products:
 Square of a Binomial:
(𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2
(𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2
 Square of a Trinomial
(𝑎 + 𝑏 + 𝑐 )2 = 𝑎2 + 𝑏2 + 𝑐 2 + 2𝑎𝑏 + 2𝑎𝑐 + 2𝑏𝑐
 Sum and Difference Formula
(𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2
 Cube of a Binomial
(𝒂 + 𝒃)𝟑 = 𝒂𝟑 + 𝟑𝒂𝟐 𝒃 + 𝟑𝒂𝒃𝟐 + 𝒃𝟑
(𝒂 − 𝒃)𝟑 = 𝒂𝟑 − 𝟑𝒂𝟐 𝒃 + 𝟑𝒂𝒃𝟐 − 𝒃𝟑

III. Assessment
A. Choose the letter of the correct answer. Write your answers on your answer sheet.

1. The expression (𝑟 + 𝑦)2 is an example of _________.

A. Square of a binomial C. Cube of a binomial
B. Square of a trinomial D. Sum and Difference of two terms
2 2
2. The expression (2𝑥 + 3𝑦)(2𝑥 − 3𝑦) is an example of _________.
A. Square of a binomial C. Cube of a binomial
B. Square of a trinomial D. Sum and Difference of two terms
3. The expression (𝑥 + 𝑦)3 is an example of _________.
A. Square of a binomial C. Cube of a binomial
B. Square of a trinomial 8 D. Sum and Difference of two terms
 4. The expression (𝑥 + 2𝑦)3 is an example of _________.
A. Square of a binomial C. Cube of a binomial
B. Square of a trinomial D. Sum and Difference of two terms
2
5. The expression (𝑥 + 2𝑦 + 3) is an example of _________.
A. Square of a binomial C. Cube of a binomial 1)
B. Square of a trinomial D. Sum and Difference of two terms

Multiply the following polynomials.

6. −7𝑥(11𝑥 + 12)
A. −77𝑥2 − 84𝑥 B. −161𝑥2 C. 11𝑥 2 − 84𝑥 D. −77𝑥2 + 12𝑥
5 2
7. −5𝑥 (10𝑥 + 5)
A. −50𝑥7 + 5 B. −50𝑥7 − 25𝑥5 C. −75𝑥5 D. −50𝑥2 – 25
3 7
8. −2𝑥 (−2𝑥 − 7𝑥 ) 2

A. 4𝑥 10 − 7𝑥 2 B. 18𝑥 10 + 18𝑥 5 C.18𝑥 3 D. 4𝑥 10 + 14𝑥 5

2
9. (4𝑚 + 9)(−4𝑚 + 𝑚 + 3)
A. −16𝑚3 + 21𝑚 + 27 C. −16𝑚3 + 40𝑚2 + 21𝑚 + 27
3 2
B. −16𝑚 − 32𝑚 + 21𝑚 + 27 D. −16𝑚3 − 40𝑚2 + 21𝑚 + 27
2
10. (9𝑦 − 1)(6𝑦 − 𝑦 − 4)
A. −3𝑦2 − 35𝑦 + 4 C.54𝑦 3 + 3𝑦 2 − 35𝑦 + 4
3 2
B. 54𝑦 − 15𝑦 − 35𝑦 + 4 3
D. 54𝑦 − 35𝑦 + 4

Multiply the binomials using FOIL.

11. (2𝑥 − 6)(𝑥 + 11)
A. 𝑥 2 − 66𝑥 + 16 C. 2𝑥 2 + 16𝑥 – 66
B. B. 𝑥 2 + 16𝑥 + 11 D. 2𝑥 2 + 11𝑥 − 66
12. (3𝑥 − 5)(𝑥 − 10)
A. 𝑥 2 − 35𝑥 – 4 C. 𝑥 2 + 50𝑥 – 35
B. B. 3𝑥 2 − 4𝑥 + 50 D. 3𝑥 2 − 35𝑥 + 50
13. (3𝑥 + 6)(𝑥 + 7)
A. 3𝑥 2 + 27𝑥 + 42 C. 𝑥 2 + 27𝑥 + 34
B. 𝑥 2 + 42𝑥 + 27 2
D. 3𝑥 + 34𝑥 + 42

14. (𝑥 + 1)(−5𝑥 + 10)

A. −5𝑥2 + 10𝑥 + 5 C. −5𝑥2 + 3𝑥 + 𝐼𝑂
B. −5𝑥2 + 5𝑥 + 5 D. −5𝑥2 + 5𝑥 + 𝐼𝑂

15. (𝑥 + 12𝑦)(𝑥 − 6𝑦)

A. 𝑥 2 + 3𝑥𝑦 − 72𝑦 2 C. 𝑥 2 + 6𝑥𝑦 + 6𝑦 2
B. 𝑥 + 6𝑥𝑦 − 72𝑦 D. 𝑥 2 + 6𝑥𝑦 − 72𝑦 2

Multiply the polynomials using the appropriate special products.

16. (2𝑞 − 4)2
17. (7𝑞 2 𝑤 2 − 4𝑤 2 )2
18. (𝑒 + 2𝑐 + 𝑞)2
19. (4𝑦 − 5𝑑)(4𝑦 + 5𝑑)
20. (2𝑥 − 𝑦)3

IV. References

1. Louis Leithold, College Algebra and Trigonometry: Pearson Education Asia Pte Ltd
2002, 29-33

Prepared by:
JASTINE MAR T. RAMOS
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 Special Science Teacher I

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