set of real numbers_edition2a_4m.pdf

7/27/2019 Set of Real Numbers_edition2a_4m.pdf

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The Set of Real Numbers

1. IntroductionThis chapter mainly talks about the set of real numbers and the operations

involving them. It includes a review discussion of the properties of real numbers,the laws of exponents, polynomials, special products, the Binomial Theorem, andfactoring of polynomials.

2. The Set of Real NumbersEveryday situations reveal that the set of real numbers is very important,

specifically the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This is known as the

number system in base ten. This was developed by the Hindus and, later, wastranslated to Arabic. There are other number systems which are of major

importance. Three of these are the base two system, the base eight system, and the

base sixteen system. In base ten system, a number may be used singly or incombination to indicate (a) quantitythere are three sides in a triangle; (b) order

I camefirst. You camesecond; (c) constructionthe numbers 1 and 2 may be

used to form any of the numbers 12, 21, 0.12, 0.21 (the zero is used to indicatethat the number is less than 1); (d)place0 is used to established the place value

as used in the numbers 60, 703, and 0.0009.

In another manner, the significance of base ten can be demonstrated by thefollowing examples:

4728 3 2 1 04 10 7 10 2 10 8 10

0.5163 1 2 3 45 1 6 3

10 10 10 10

3. Subsets of Real NumbersThe set of real numbers has many important subsets. Some of these

subsets are enumerated below:

Natural NumbersThe set of natural numbers is also called the set of positive integers. This

set is also known as the counting numbers. This set is denoted by , and may

be written as 1, 2, 3, ...

Whole NumbersThe set of whole numbers is composed of the set of natural numbers and

the number zero, 0, and is denoted by .W By set notation, 0W which

may be written as

0, 1, 2, 3, ... .W


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Negative Integers

The set of negative numbers, or the set of all negative n for all ,n arouse to allow solutions to the equation ,x a b where , .a b Equivalently,

for each positive integer ,n there exists a negative integer such that if it is added

to ,n the sum is 0. For example, if 10 is added to 10, then the sum is equal to 0.

The negative numbers, denoted by , may be written as

..., 3, 2, 1 .

Integers

The set of integers is the union of three disjoint sets: the positive integers,

zero, and the negative integers, and denoted by , is given by

..., 3, 2, 1, 0, 1, 2, 3, ...

Rational Numbers

These numbers are commonly called fractions. Rational numbers arouse to

allow solutions to the equation bx a for all , ,a b and 0.b The set of

rational numbers is denoted by .

When 1,b x a . That is, .x Hence, .

Some numbers in decimal form are considered rational. These numbers

may be either repeating or terminating decimals. For example, consider the

following rational numbers with their corresponding decimal form:1

2 0.5,

25

90 0.2777...,

357

999 0.357357...,

817

900 0.90777...

Irrational Numbers

When a numberx cannot be considered as a rational number, then it iscalled an irrational number. This means that no fraction corresponds tox.

Examples of this set of numbers are e, , and 0.2314527... . The set of irrational

numbers is denoted by '.

In the examples, the first and the second of the numbers have infinitely

many digits that are not repeating when written in decimal form. With the thirdnumber, the digits before the three dots did not repeat. Thus, these numbers arenonrepeating, nonterminating decimals.

When the rational and the irrational numbers are combined, the set of real

numbers will be formed. Thus, .


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Figure 1.1 (a to c) shows different representations of the set of real

number. It also shows the subsets of and their relationship with each other.

Other subsets of such as the odd numbers, the even numbers, the prime

numbers, the multiples of 2, multiples of 3, etc may be added.

4. Properties of Real NumbersThe Real Number System is composed of , together with the binary

operationaddition and multiplication or . This enables us toconclude that the system forms a field.

Suppose , , ,a b c we define the field axioms.

a. ,a b ab Closure Lawb. ;a b b a a b b a Commutative Lawc. ;a b c a b c Associative Law

a b c a b c

d. ;a b c a b a c (Right)Distributive Law a b c a c b c (Left)Distributive Law

e. 0 0 ;a a a 1 1a a a Identity Lawf. 0;a a 1 1aa Inverse Law

The 0 and the 1 in and (e) and (f) are called the additive and

multiplicative identity elements, respectively. To further elaborate lawf, consider

the following arguments:

For any ,a there exists a b such that 0.a b For any ,a there exists a b such that 1.ab

In the first statement, b is commonly known as the opposite. In the

second statement, b is commonly known as the reciprocal.

W

0

W

W

Figure 1.1a Figure 1.1b Figure 1.1c


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5. Geometric Representation of Real NumbersOften, real numbers can be represented by points on a line. This is called

the real number line. This permits us to speak of the set of points rather than the

set of real numbers, hence, the statement: For each real number, there corresponds

one and only one point on the line and vice versa. This can easily be understoodwith Figure 2.

In the number line, observe that between two rational (irrational) numbersthere are infinitely many rational and irrational numbers. This leads to the idea

that the set of real numbers is dense everywhere.

6. Operations Involving Real NumbersThere are two predefined operations involving the set of real numbers:

addition and multiplication. These operations were discussed below.

In adding two numbers with like signs, find the sum of their absolute

values and prefix the common sign. For example, 3 5 8; 11 8 20. In adding two numbers with unlike signs, find the difference between their

absolute values and prefix the sign of the number with greater absolute value. For

example, 17 6 11;

11 8 3.

In multiplying two numbers with like signs, find the product of their

absolute values. For example, 13 6 78; 21 7 147. In multiplying twonumbers with unlike signs, find the product of their absolute values and prefix a

minus sign. For example, 9 6 54; 8 9 72.

For simplicity, the product of a and b can be written as .ab Also, two

operationssubtraction and divisionwere considered when involving . They

are defined as a b a b and1

,a

a

b b

, ,a b where b and1

b

are

the opposite and the reciprocal ofb, respectively, that follows from the previous

operations. Thus, 13 19 13 19 6; 17 28 17 28 11;

57 1

57 3;19 19

72 1 9

72 4.5.16 16 2

102 3

1

2 10

3

6

Figure 2. The Real Number Line


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Seat Work 3

Name: Score:

Program, Year & Section: Date:

A. Determine whether each statement is true or false._________ 1. Every integer is a rational number._________ 2. Some real numbers are integers._________ 3. If ,x then x can be both a rational and an irrational._________ 4. A real number can be both an integer and a rational._________ 5. There are as many real numbers between 0 and 1 than between 1 and 100.B. Determine the properties involved in every statement. Assume that , , .a b c ___________________ 1. 2 7 2 14a a ___________________ 2. 3 3c a c a ___________________ 3. 2 2 0a b a b ___________________ 4. 122 1abab ___________________ 5. ac bc a b c C. Perform the indicated operations.

1. 2 3 7 1 2. 5 2 7 3. 6 1 7 3 4. 4 2 3 5. 11 7 9 15 6. 1 4 3 2 7. 5 3 1 2 1 8. 1 2 1 2 9. 72 2 3 12 10. 36 1 3 4


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D. Discuss each of the following statements.1. The set of natural numbers is closed under division.

2. Some counting numbers are fractions.

E. Classify each of the following numbers according to categories by completing thetable below. Use a check mark if the number belongs to the set.

None of the

given sets

1. 3.25

2. 3

3. 0.0000025

4. 0

5. 27

6. 4

7. 9

8. 1123

9. 0.809523

10. 8


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7. Properties of EqualityLet , , .a b c The following are equivalence properties:

a. Reflexive property: .a a b. Symmetric property: If ,a b then .b a c. Transitive property: If a b and ,b c then .a c d. Addition property: If ,a b then .a c b c e. Multiplication property: If ,a b then .ac bc f. Substitution property: If ,a b then you may consider b in

place of .a

To include subtraction and division, we have

g. Subtraction property: If ,a b then .a c b c h. Division property: If a b and 0,c then .a b

c c

8. Properties of ZeroLet .a The following are the properties of zero:

a. 0 ;a a 0 .a a 8 0 0; 8 0 8. b. 0 0.a 12 0 0. c. 0 0,

a 0.a

00

132

d. ,0

aundefined 0.a

3

0undefined

e. If 0,ab then either 0a or 0b or 0.a b This is known as theZeroProduct Property.

f. 0 1,a 0.a 0123456789 1 g. 00 ,indeterminate 0 .

0indeterminate

9. Order of OperationsIn any case, the convention PEMDAS Parentheses, Exponent,

Multiplication, Division, Addition, and Subtraction will be used. It is important tonote that the parenthesis or any other symbols of groupings comes first before

other operations. Then comes the exponents, and so on. For example, consider

2 2

3 8 4 7 4 3 5 4 7 12 25 4 5 25 20 5.

When no groupings are involved, this convention goes down to

EMDAS. This does not imply that multiplication should be used first before

division. It follows theFirst In,First Out Principle. The same is the case withaddition and subtraction. As such,

2 39 3 4 3 6 2 27 16 3 6 8 27 32 8 3.


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10.Symbols of GroupingsOften it is desired to group two or more terms to indicate that they are to

be considered first or to be treated as though they were one term, even though it

may contain other operations. Thus, symbols of grouping were presented below.

parentheses braces

brackets aaa vinculum

11.The Laws of ExponentsLet .a The product of a n times to itself and written as

n

a a a a

is denoted by ,na where a is called the base and n is called the exponent. For

instance, if there is anNsuch that na N then N is called the nth power of a .

Suppose , .m n Then the following holds true for .

a. .m n m na a a 4 7 115 5 5 ; b. , 0.m m n

n

aa a

a

7

4

3

22 ;

2

c. .nm mna a 23 64 4 ; d. .n n nab a b 33 2 6 312 2 3 2 3 ; e. , 0.n n

n

a ab

b b

7 7

7

2 2;

3 3

In ,b m n leads to 0 1a while 0m results to1

.nn

aa

12.PolynomialsApolynomialis an algebraic expression composed of one or more terms.

A term in a polynomial may consists of ordinary numbers and/or letters which

represents numbers. These numbers and letters are interrelated by addition,

subtraction, and multiplication. This implies that no letters appear in the

denominator. Thus, 213ab c and 3 24x y may be considered as terms in a

polynomial.

In a term, the number is known as the numerical coefficient (or simplycoefficient) while the letters is commonly known as the literal coefficients (or

simply variables). For instance, in 4 36 ,s t 6 is the numerical coefficient while


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4 3s t is the literal coefficient. If a term has no variables, then it is said to be a

constant. Otherwise, it tends to vary given arbitrary values of the variables.

Two or more terms that differ only in their numerical coefficients are said

to besimilar terms; otherwise, they are dissimilar terms. Thus, 32x y and 37x y

are similar terms while 5xy and 28xy are dissimilar terms.

The degree of a term is determined by the sum of all the exponents of the

variables present. Thus, the degree of 3 2 52x y z is 3 2 5 10.

The degree of a polynomialis defined as the highest degree of the terms in

the polynomial. For instance, the polynomial 3 3 4 2 43 7 4xy x y z xy z is of degree

9 because the second term has the degree 3 4 2 9 and is the highest of the

three degrees. The first and the third terms are only of degrees 4 and 6,respectively.

A polynomial may be classified according to the number of terms it has.

These are: monomialone term; binomialtwo terms; trinomialthree terms;

and multinomialfour or more terms. As such, 2xy is a monomial; 311 7a ab

is a binomial; 3 4m n p is a trinomial; and 4 8 3 9a b c is a multinomial.

When the terms of a polynomial is written such that the exponents of the

variable is decreasing from left to right, then the coefficient of the first term is

called the leading coefficient. For example,3

5 3 1x x can be written as

33 5 1x x and 3 is the leading coefficient.

13.Elementary Operations Involving PolynomialsOccasionally, two or more polynomials were combined to form another

polynomial. The combinations of these polynomials may be done by addition,subtraction, multiplication, and division.

Sum of Polynomials. In adding polynomials, combine similar terms by

adding their numerical coefficients. This process follows .ad bd a b d For

example, 2 2 2 23 8 3 8 11 .x y x y x y x y Also, 3 8 3 8a b a b because theterms are dissimilar.

When symbols of groupings are involved where a precedes a symbol ofgroupings, then this symbol of groupings may be removed without affecting the

signs contained terms.


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For example,

2 25 6 3 2 7 6x x x x 2 25 6 3 2 7 6x x x x 2 25 2 6 7 3 6x x x x

25 2 6 7 3 6x x

27 3.x x

Difference of Polynomials. In subtracting polynomials, combine similar

terms by subtracting their numerical coefficients. This process follows

.ad bd a b d For example, 4 4 4 412 7 12 7 5 .pq pq pq pq Also,2 2

5 3 5 3m n m n because the terms are dissimilar.

When symbols of groupings are involved where a precedes a symbol ofgroupings, then this symbol of groupings may be removed if the sign of each termcontained is changed. For example,

2 29 4 6 7 5 11y y y y 2 29 4 6 7 5 11y y y y 2 29 7 4 5 6 11y y y y

29 7 4 5 6 11y y

22 9 17.y y

Product of Polynomials. Multiplication of polynomials can be attained by

multiplying the terms of the factors. Generally, this follows the distributiveproperty.

When multiplying two monomials: Apply the laws of exponents and therules of sign. For example,

7 3 2 4 3 8 6 53 4w x y z w x y z 7 3 3 8 2 6 4 53 4 w x y z 10 11 8 912 .w x y z

When multiplying a monomial to a polynomial: Multiply the monomial to

each term of the polynomial. For example,

2 3 3 4 33 2 7a b a b a b 2 3 3 2 3 4 33 2 3 7a b a b a b a b

2 3 3 1 2 4 3 36 21a b a b

5 4 6 66 21 .a b a b

When multiplying a polynomial to another polynomial: Multiply eachterm of the first polynomial by each term of the second polynomial.


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For example,

3 4 2 5a b p q 3 2 3 5 4 2 4 5a p a q b p b q 6 15 8 20 .ap aq bp bq

This product may also be accomplished by considering the firstpolynomial as a monomial and follows the second concept.

3 4 2 5a b p q 3 4 2 3 4 5a b p a b q

3 2 4 2 3 5 4 5a p b p a q b q 6 8 15 20 .ap bp aq bq

Still, this product may be accomplished by aligning the polynomials

vertically and follows a procedure similar to multiplying two or more digitnumbers and combine similar terms.

3 42 5

6 8

15 20

6 8 15 20

a bp q

ap bp

aq bq

ap bp aq bq

Similarly, it is better to arrange the terms with decreasing exponents. The

product 2 3 23 4 2 4x x x x can be done as3 2

2

5 4 2

4 3

3 2

5 4 3 2

3 4

4 24 12 16

3 4

2 6 8

4 11 10 4 8

x x

x xx x x

x x x

x x

x x x x x

Division of Polynomials. The division of polynomials generally follows

the laws of exponents. Two cases will be discussed: polynomial divide by a

monomial, and polynomial divided by polynomial.

When dividing a polynomial by a monomial, divide each term of the

polynomial by the monomial. For example,7 4 3

2

12 18 6

3

x x x

x

7 4 3

2 2 2

12 18 6

3 3 3

x x x

x x x

5 24 6 2 .x x x


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When dividing a polynomial by a polynomial, write the dividend into the

form with decreasing exponents from left to right. Then, the long divisionalgorithm that was introduced in arithmetic can be used. For example,

2

3 2

3 2

2

2

00 0000

0

3 4

2 5 10 8

2

3 10 0

00 000

00 000

00 000

00

0

3 6

4 8

4 8

0000 4

x x

x x x x

x x

x x

x x

x

x

x

Thus,

3 225 10 8

3 4.2

x x x

x xx

In the next example, you will see that the set of polynomials is not closedunder division. That is, a polynomial divided by a polynomial may not always

results to a polynomial. Consider the division below.2

3 2

3 2

2

2

2

3 2 2

3

00 00

00 0

00 0

00

00 000

00

3

2 2

2000

00 000 4

6

8

x x

x x x x

x x

x

x x

x x

x

x

Thus,3 2

22 2 82 .3 3

x x xx x

x x

To check,2

3 2

2

3 2

2

3

2

3 3 6

2 6

x x

x

x x x

x x

x x x

and 3 2 3 22 6 8 2 2x x x x x x which results to the dividend.


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Seat Work 4

Name: Score:


A. Determine the degree n of each of the following polynomial.1. 2 3 4 25 9 7;s t u v n 2. 9 7 5 6 7 4 3 11 5 87 11 21 ;a m p u v x z b c d n 3. 37; n 4. 4 33 12 235 19 ;s t u v n 5. 2 13 5 120 ;c e k m n

B. Classify each of the following expressions according to categories by completingthe table below. Use a check mark if the number belongs to the set.

Monomial Binomial Trinomial Multinomial Polynomial

1. 32

5a b

cd

2. 23 7 3m m 3. 7 3 2 5k k 4. 2 28 6 5 3d e d e e 5. 12 6. 23 4 5x y z 7. 5 4 5a b a b 8. 2 2 2x y z

a b c

9. 13

x

y

10. 3 38a


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C. Remove the symbols of grouping in each of the following polynomials andsimplify the resulting expression.

1. 7 4 9 ;m m n 2. 5 4 4 5 ;h h j 3. 2 2 2 2 25 11 6 7 8 ;a b a b ab ab a b 4. 2 3 2 3 4 ;r s r s r s r s 5. 3 4 5 4 2 3 5 2 3 ;de f d de f de

D. Evaluate the following powers and express the result without zero or negativeexponents.1. 4 7 9 132 16 ;d e d e 2. 4 8 2 3 11 1

1 6 10 7 5 9;

x y z x y z

x y z x y z

3. 0

3 4 13 4

5 3 4 5

2

2;

m nm n m n

p

4. 2 4 33 4 3 5 4 52 ;a b a b a b 5.

213 4 3 1

4 2 63 5

;s t s t

s ts t

E. Perform the indicated operations and simplify.1. 5 3 5 4 33 4 2 5 ;m m m m m m m 2. 2 25 4 3 7 3 4 ; 3.

23 2 6 3 5 ;

4. 3 3 2 4 2 5 4 32

75 25 50 15;

5

5. 3 24 8 11 18 ;2 3


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14.Special ProductsSome products occur in mathematics that it is advantageous to recognize

them at once. These products are enumerated below.

I.

a b c d ab ac ad

II. a b a b 2 22a ab b III. a b a b 2 2a b IV. x a x b 2x a b x ab V. ax by cx dy 2 2acx ad bc xy bdy VI.

3a b 3 2 2 33 3a a b ab b

VII. 2a b c 2 2 2 2 2 2a b c ab ac bc Lets solve several examples for each of these products.

a. 2 33 4 9s r s t rst 2 33 4 3 3 9s r s s t s rst 2 4 212 3 27r s s t rs t

b. 3 2 4 3 4 25 2 7 3x y x y xy x y 3 2 4 3 3 3 4 25 2 5 7 5 3x y x y x y xy x y x y

5 5 4 4 7 3

10 35 15x y x y x y

c. 26 5d 226 2 6 5 5d d 236 60 25d d

d. 223 7mn p q 22 2 23 2 3 7 7mn mn p q p q 2 2 2 4 29 42 49m n mnp q p q

e. 2 11 2 11j k j k

2 22 11j k

2 24 121j k

f. 3 4 2 5 3 4 2 52 5 2 5a b c m np a b c m np 2 23 4 2 52 5a b c m np 6 8 2 4 2 104 25a b c m n p


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g. 13 12x x 2 13 12 13 12x x 2 156x x

h. 5 3 5 7abx cd abx cd 25 3 7 5 3 7abx cd cd abx cd cd 2 2 2 2 225 50 21a b x abcdx c d

i. 4 5 3 2x y x y 2 24 3 4 2 5 3 5 2x xy y 2 212 7 10x xy y

j. 2 210 3 9 5

22 2 210 9 10 5 3 9 3 5

2 2 2 4 290 23 15

k. 33 2a b 3 2 2 33 3 3 2 3 3 2 2a a b a b b 3 2 2 327 54 36 8a a b ab b

l. 32 3 4 29 11w v y z

3 2 2 32 3 2 3 4 2 2 3 4 2 4 29 3 9 11 3 9 11 11w v w v y z w v y z y z

6 9 4 6 4 2 2 3 8 4 12 6729 2673 3267 1331w v w v y z w v y z y z

m. 22 3 5m n p

2 2 22 3 5 2 2 3 2 2 5 2 3 5m n p m n m p n p

2 2 24 9 25 12 20 30m n p mn mp np

or

2

2 3 5m n p

2 2

2 3 2 2 3 5 5m n m n p p

2 2 24 12 9 20 30 25m mn n mp np p

n. 24 5 3 2a b c d 24 5 3 2a b c d

2 24 5 2 4 5 3 2 3 2a b a b c d c d

2 2 2 216 40 25 24 16 30 20 9 12 4a ab b ac ad bc bd c cd d


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Seat Work 5

Name: Score:


A. Find each of the following products.1. 2 2 3 3 2 43 5 2 8x y x y x y xy

2. 223x y

3. 23 2 7 57 9m x n y

4. 2 3 4 6 2 3 4 64 5 4 5a b b c a b b c

5. 2 2 25 2 3 4u vw v u w

6. xy yz xz xy yz xz

7. 2 2 2 2x xy y x xy y


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8. 22 2a bc d

9. 32 2x y xy

10. 23 23 4j kl m n

B. Use the special products1. 12 13 21 12 12 18

2. 2498

3. 479 521

4. 689 531 689 376 689 93

5. 2 2362 638


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15.Binomial TheoremThe product of a binomial n times to itself written as

na b follows a

certain pattern. For instance,

0

a b 1

1

a b a b

2

a b 2 22a ab b

3

a b 3 2 2 33 3a a b ab b

4

a b 4 3 2 2 3 44 6 4 , .a a b a b ab b etc

With this, the expansion n

a b can be summarized by Definition 6.

With Definition 6, the binomial expansion n

a b has several properties.

These are the following:

a. There are 1n terms in the expansion.b. The first term a appears in the first term with exponent ,n in the second

term with exponent 1 ,n in the third term with exponent 2 ,n and

in the thr term with exponent 1 .n r

c. The second term b appears in the second term with exponent 1, in thethird term with exponent 2, in the fourth term with exponent 3, in the thr

term with exponent 1 ,r and in the last term with exponent .n

d. The sum of the exponents of a and b in each term of the expansion isequal to .n

e. The coefficient of the first term in the expansion is 1, of the second term is,

1

nof the third term is

1,

1 2

n n

of the fourth term is 1 2

,1 2 3

n n n

and so on. The coefficient of any term can be computed using combination

and is given bythr term 1 11

n r rn rC a b

1 11n n r r r a b

De inition 6. The Binomial Theorem

n

a b 1 2 2 2 21 1 21 2 1 2 3

n n n nn n n n na na b a b a b

1 1 11 2 2

1 !

n r r n nn n n n r a b nab br


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where 1n rC !

1 ! 1 !

n

n r r

1 2 2.

1 !

n n n n r

r

f.

The coefficients of terms equidistant from the ends of the expansion arethe same.

g. There are two middle terms when n is odd.h. There is only one middle term when n is even.

In the expansion of 10

2 3,r s let 2 ,a r 3,b s and 10.n Then the

expansion is

10

2 3r s 10 9 1 8 2 7 3

2 10 2 3 10 2 3 10 2 31 2 3r r s r s r s

6 4 5 5 4 6

10 2 3 10 2 3 10 2 34 5 6r s r s r s

3 7 2 8 1 9 10

10 2 3 10 2 3 10 2 3 37 8 9r s r s r s s

10

2 3r s 20 18 3 16 6 14 9 12 12 10 1510 45 120 210 252r r s r s r s r s r s

8 18 6 21 4 24 2 27 30210 120 45 10r s r s r s r s s

Consider another example, the expansion of 6

3 4 .p q Let 3 ,a p

4 ,b q and 6.n Then the expansion is

6

3 4p q 6 5 1 4 2 3 36 6 6

1 2 33 3 4 3 4 3 4p p q p q p q

2 4 1 5 66 6 6

4 5 63 4 3 4 4p q p q q

6

3 4p q 6 5 4 2 3 3 2 4729 5832 19440 34560 34560p p q p q p q p q

5 618432 4096pq q

Now, suppose only certain terms are required such as the eighth and the

middle terms of 10

9 4 .h k Here, 9 ,a h 4 ,b k and 10.n The middle term

is the 2 1th

n term when n is even, and the 12th

n and 12 1th

n terms when n

is odd. Thus, the middle term is 102 1 6th th term. Hence,

8th

term 10 8 1 8 110 8 1C a b

3 7

10 9 47 h k

27 28120 .h k

6th

term 10 6 1 6 110 6 1C a b

5 5

10 9 45 h k

45 20252 .h k


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Seat Work 6

Name: Score:


A. Expand following using binomial theorem and simplify.1. 32 3 45x y z

2. 53 5 3 22m n p q

3. 42 31 24 3x y yz

4. 563 5e fg

5. 35 3 2 7 3r s t u


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B. Find the indicated term and simplify.1. Fifth term of 75 33 4c d

2. Third term of 105 33ac d g

3. Fourteenth term of 1522xy wz

4. Middle term of 64 33f wx

5. Middle terms of 95 2 3 623 h k m p

C. Write the first five terms of the following expansion.1. 30y z

2. 203 24 3a b


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16.Factoring PolynomialsFactoring process for polynomials is finding factors such that when

multiplied together will result to the given polynomial.

The factors required here should be primes, that is, the obtained factorscannot be factored according to the following restrictions:

a. Factors are said to be prime if its factors are only 1 and plus or minus ofitself. This is in accordance with the definition of prime numbers such as

2, 3, 5, and 7.

b. All factors of polynomials should also be polynomials with integercoefficients. For instance, 344 3 4x x shall not be considered as

factored form. Also, 4 2 2x x x is not allowed because thefactors are not polynomials.

c. Trivial changes in sign shall be allowed. Thus, 2 4 3x x can be factoredas either 1 3x x or 1 3 .x x

d. Many cases require us to factor polynomials with rational coefficients. Asan example, 2 29 9 3 31 1 12 8 2 4 2 2 2y y y y may be allowed inthese cases. Otherwise, we adhere to the previous limitations.

17.Factoring MethodsFactoring expressions such as polynomials and rational expressions

follows certain procedures to facilitate the factoring process. Previous discussionsin multiplication will be of great help here. Thus, there is a need to master thembefore going to this topic.

The following procedures are very useful.a. Greatest Common Factor. This method factors the greatest common factor

from each term, thereby simplifying the given expression.

ab ac ad a b c d where .GCF a

i.e. 2 3 4 23 6x y x y 2 2 23 2 .x y y x

b. Difference of two squares. The given expression is the difference of twosquares and the factors are the sum and the difference of two terms.2 2a b a b a b

i.e. 2 4 69 16x y w 2 2

2 33 4xy w 2 3 2 33 4 3 4 .xy w xy w


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c. The square of a binomial. The given expression is a perfect squaretrinomial, thus, it is a square of the binomial.

2 22a ab b a b a b

2a b

i.e.2 2

4 36 81m mn n 2 2

2 2 2 9 9m m n n 2

2 9 .m n 2 2

49 84 36u uv v 2 2

7 2 7 6 6u u v v 2

7 6 .u v

2 2100 160 64f fg g 2 2

10 2 10 8 8f f g g 2

10 8 .f g

d. Other special products. The given expression follows certain patternsimilar to those that can be obtained by multiplication.

2x a b x ab x a x b

2acx ad bc x bd ax b cx d 3 2 2 3

3 3a a b ab b 3

a b

2 2 22 2 2a b c ab ac bc

2a b c

i.e. 2 26v vw w 3 2 .v w v w 2 2

2 3s st t 2 3 .s t s t 3 2 2 327 54 36 8e e f ef f

33 2 .e f

2 24 1 4 4 2p q pq p q 2

2 1 .p q

Note that it is very advantageous on your part if you can easily recognize

the pattern present in the given expression.

e. The difference or sum of two cubes. The factors of the difference of twocubes follows the form

3 3a b 2 2a b a ab b

i.e. 3 38 27v w 3 3

2 3v w 2 22 3 4 6 9 .v w v vw w 9 6 364i j k

3 33 24i j k 3 2 6 3 2 4 24 16 4 .i j k i i j k j k

Unlike the sum of two squares that is not factorable, the sum of twocubes is indeed factorable and it follows the form

3 3a b 2 2a b a ab b

i.e. 3 664m p 33 24m p 2 2 2 44 16 4 .m p m mp p

6 31000c d

3 32 10c d 2 4 2 210 10 100 .c d c c d d


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f. Grouping of terms. This is a method wherein the terms are group in such away that there is a common factor among the terms in the groups formed.This method may be best illustrated by an example.

Factor completely: 23 2 12 8x x x Applying grouping of terms,

23 2 12 8x x x 23 2 12 8x x x

3 2 4 3 2x x x 2

3 2 12 8x x x 4 3 2 .x x

Consider another example. Factor completely the following expression:

abx acx bcy aby bcx acy

Likewise, by grouping similar terms,

abx acx bcy aby bcx acy abx acx bcx aby acy bcy

ab ac bc x ab ac bc y

abx acx bcy aby bcx acy .ab ac bc x y

g. General factors of .n na b i For ,n the expression a b is a factor of .n na b That is,

1 2 3 2 2 1 .n n n n n n na b a b a a b a b ab a

i.e. 2 2a b a b a b 3 3a b 2 2a b a ab b 4 4a b 3 2 2 3a b a a b ab b 5 5a b 4 3 2 2 3 4a b a a b a b ab b

ii For 2 ,n the expression a b is a factor of .n na b That is,


i.e. 2 2a b a b a b 4 4a b 3 2 2 3a b a a b ab b 6 6a b 5 4 3 2 2 3 4 3a b a a b a b a b ab b 8 8a b 7 6 5 2 4 3 3 4 2 5 6 7a b a a b a b a b a b a b ab b


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iii For ,n Odd the expression a b is a factor of .n na b That is,


i.e. 3 3a b a b a b 5 5a b 4 3 2 2 3 4a b a a b a b ab b 7 7a b 6 5 4 2 3 3 2 4 5 6a b a a b a b a b a b ab b

iv For 2 ,n the expression a b is not a factor of .n na b However, when n has an odd numberp as a factor, and ,n pq then

n na b can be factored using .iii That is,n na b pq pqa b

p pq qa b and use .iii

i.e. 4 4a b 4 4a b because 4n has no odd number as a factor.

6 6a b 3 3

2 2a b 2 2 4 2 2 4a b a a b b 10 10

a b 5 5

2 2a b 2 2 8 6 2 4 4 2 6 8a b a a b a b a b b

h. Factoring by completing the squares. This method is only used when allof the above methods did not work. This is best explained with the

following examples.

Consider factoring completely the expression 4 44 .x y At first,

you may tell that this is not factorable. However, by this technique, it canbe shown that the given expression is definitely factorable. To show this,

4 44x y 4 2 2 4 2 24 4 4x x y y x y

2 22 22 2x y xy

2 2 2 22 2 2 2 .x xy y x xy y

In another case, factor 4 2 2 436 25x x y y completely. Similarly,

4 2 2 436 25x x y y 4 2 2 4 2 220 25 16x x y y x y

2 22 25 4x y xy

2 2 2 24 5 4 5 .x xy y x xy y


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Seat Work 7

Name: Score:


A. Factor completely the following expressions over the set of polynomials withinteger coefficients.

1. 2 5 3 3 4 215 9a b c a b c

2. 9 11 13 12 8 10 13 9 112 18 6x y z x y z x y z

3. 2 20m m

4. 23 10ax ax a

5. 2 23 18 27b bc c

6. 3 2 3 3100 80 16g m g m g

7. 2 3 48 18f g f g

8. 3 23 9 9 3b b b

9. 3 227 27 9 1x k x k x k


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10. 3 2 3 3100 80 16g m g m g

11. 42 16hx hx

12. 2 49 16p q

13. 6 63 81r s

14. 664 486kx y kxy

15. 6 52 2x x x

16. 3 22 4 2 4i i i

17. 4 2 2r r w r s sw

18. 664 486kx y kxy

19. 3 2 3 2 3 2 3 2a b a c d b d c

20. 4 3 23 5 6 8z z z z


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set of real numbers_edition2a_4m.pdf

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