chapter r-basic concepts of algebra r.1 the real number …mathlady.org/mathlady/mac1105 college...

MAC 1105-College Algebra LSCC, S. Nunamaker

Chapter R-Basic Concepts of Algebra

R.1 The Real Number System

I. Real Number System

Please indicate if each of these numbers is a W (Whole number), R (Real number), Z(Integer), I (Irrational number), N (Natural number), Q (Rational number)

Examples: -6, 7, 5 4. , -3

7, 83 ,

6242242224. , 624224222422224. ....,

P, -5

11 , 2, 193 ,

-02020020002. ....

Natural Numbers: {1, 2, 3, 4,....}

Whole Numbers: {0, 1, 2, 3, 4, ....}

Integers: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ....}

Rational Numbers: {p

qp and q are integers and q ¹ 0}

Irrational Numbers: {x x is real but not rational}

Real Numbers: {xx corresponds to a point on a number line}

II. Properties of Real Numbers For any real numbers a, b, and c:

a + b = b + a

a b = b a

a + ( b + c ) = ( a + b ) + c associative property

a ( b c ) = ( a b ) c associative property

a + 0 = 0 + a = a identity property

a -a = a + ( - a ) = 0 inverse property of addition

1

a × 1 = 1× a = a identity property LSCC,S. Nunamaker

aa

×1

= 1 such that a ¹ 0 inverse property of multiplication

a ( b + c ) = a b + a c distributive property

Examples: 1. 3 2 2 3 5+ = + = 8. 101

101× =

2. ( )( ) ( )( )3 2 2 3 6= = 9. 2 6 4 12 8 20( )+ = + =

3. 2 3 4 2 3 4 9+ + = + + =( ) ( )

4. 2 3 4 2 3 4 24( ) ( )× = × =

5. 6 0 0 6 6+ = + =

6. 6 6 6 6 0- = + - =( )

7. 4 1 1 4 4× = × =

III. Absolute Values: Distance on the number line from 0 to the number. Absolute value isalways positive. Absolute value of a is a

Example: 7 7= - =7 7

IV. Distance

Absolute value of the difference between two numbers

Examples: a. distance between -2 and -8

b. distance between - 4 and -14

c. distance between 12.1 and 5.1

d. distance between 15

8 and

23

12

e. distance between 16 and -8

2

R.2 Integer Exponents, Scientific Notations, and Order of Operations LSCC,S. Nunamaker

I. Integers as Exponents

A. For any positive integer n, a a a a a an = × × × ××× n times

such that a is the base and n is the exponent

Example: a a a a3 = × × 3 3 3 33 = × × = 27

B. For any nonzero real number a and any integer n,

a0 1= and aa

n

n

- =1

Examples: 1. 60 =

2. ( )- =2 0

3. ( )- =2 3

4. -23 =

5. 4 2- =

II. Properties of Exponents

a a am n m n× = + ( )ab a bm m m=

( )a

b

a

bm

m

m= such that b ¹ 0

a

aa

m

n

m n= -( ) such that a ¹ 0

( )a am n mn=

Examples:

1. ( )2 2 5x - = 2. 45

15

8

2

x

x=

3. y y- × =5 2 4. x x2 5× =

5. y y- -¸ =6 3 6. y

y

2

4-=

3

7. ( )- =2 3 4x 8. y

y

-

=3

2 LSCC, S. Nunamaker

9. m m- × =5 5 10. ( )2 3 5x - =

11. ( )24

3

10 8 7

6 3 5

5a b c

a b c

-

-= 12. ( )

27

9

4 2

2 8

3x y

x z

-

-

- =

III. Scientific Notations

Scientific notation for a number is an expression of the type

N x 10m , such that 1 10£ <N , N is in decimal notation, m is an integer

In scientific notation, numbers appear as a number greater than or equal to 1 and less than10 multiplied by some power of 10.

Ex. 1. FM radio signal may be 14,200,000,000 hertz (cycles per second); in scientificnotation, this is 1.42 ́ 1010 hertz.

2. Diameter of an atom is 0.0000000001 meter,

In scientific notation, this is 1 10 10´ - meter

Try to express the result in scientific notation:

1. ( . )( . )91 10 82 1017 3´ ´ =- 2. 6 4 10

80 10

7

6

.

.

´

´

-

=

3. 145 000 000, , = 4. 000876. =

Try to express the following without exponents:

1. 23 104. ´ = 2. 897 10 5. ´ - =

3. 146 106. ´ = 4. 457 10 3. ´ - =

*In 2000, the number of people living in the world was about 609 109. ´ . The number of peoplelived in US at that time was about 2 74 108. ´ . How many people lived outside of US in 2000?

*We have proof that there are at least 1 sextillion, 1021 , stars in the Milky Way. Write thisnumber without the use of exponent.

4

LSCC, S. Nunamaker

Examples:

1. Convert each of the followings to decimal notation:

a. 7654 10 5. ´ - = b. 6 45 105. ´ =

2. Convert each of the followings to scientific notation:

a. 876 420 000, , = b. 0000542. =

c. ( . )( . )91 10 82 1017 3´ ´ =- d. 6 4 10

80 10

7

6

.

.

´

´

-

=

e. 432 3 4 1012. ( . )´ ´ = f. 25 10

5 10

6

2

. ´

´=

g. 46 26 10

25 4 10

8

2

. .

.

´ ´

´ ´=

-

IV. Order of Operations

Please Excuse My Dear Aunt Sally

( ) a exp ́ ̧ + -

1. Exponential expression and calculations within grouping

symbols first and always from left to right.

2. Multiplication and division from left to right

3. Addition and subtraction from left to right

Examples:

1. ( )6 3 22 + = 2. 8 5 3 103( )- - =

3. - - - +4 3 2 42 2 2{ [ ( )]}a a b a 4. [ ( ) ]( )

( )

4 8 6 4 3 2 8

2 2 5

2

2 3

- + - ×

+=

5. 2 3 5 2 22b b b b- + -[ ( )] = 6. 4 8 6 4 3 2 8

3 19

2

1 0

( )- - × + ×

+ =

5

V. Compound Interest Formula LSCC, S. Nunamaker

Principal P is invested at an interest rate i, compounded n

times per year, in t years it will grow to an amount A given

by: A Pi

nnt= +( )1 or A P

r

nnt= +( )1

A: total amount, P: principal, i r= = annual interest rate

n: number of times compounded per year, t: number of years

Examples:

1. Suppose $9550 is invested at 5.4%, compounded semiannually. How much is in theaccount at the end of 7 years?

2. Suppose $6700 is invested at 4.5%, compounded quarterly. How much is in theaccount at the end of 6 years?

3. Suppose you will be needing $20,000 in ten years, how much do you need to investnow (principal), if the investment will be earning interest at 10% and compounding semi-annually?

4. Suppose you will be needing $20,000 in ten years, how much do you need to investnow {principal}, if the investment will be earning interest at 10% and compounding quarterly.

5. Suppose you will be needing $20,000 in ten years, how much do you need to investnow (principal), if the investment will be earning interest at 5% and compouding semi-annually.

6. The interest earned on an $800 investment at 71

4% annual interest compounded

monthly for 6 months is ?

7. Shane begins a new job with an annual salary of $40,000 and a guarantee of a 2%salary increase every year for the first 5 years. After that, he is given 3% increase each year. What will be his salary in 8 years?

8. A couple want to have $100,000 in 21 years for their newborn son. How much money should be deposited now in an account earning 6

1

2% annual interest compounded quarterly?

6

R.3 Addition, Subtraction, and Multiplication of Polynomials LSCC, S. Nunamaker

I. Polynomials in One Variable

a x a x a x a x ann

nn+ + + + +-

-1

12

21 0....

and that n is a nonnegative integer,

and a an,..., 0 are real numbers called coefficients

and an ¹ 0

A. Definition

1. Terms

2. Degree of the polynomial

3. Leading coefficient

4. Constant term

5. Descending order

Examples:

a. 2 8 204 3x x x- + - b. y y2 31

26- +

6. Monomial

7. Binomial

8. Trinomial

II. Polynomial in Several Variables

A. Definition

1. Degree of a term - sum of the exponents of the variables in that term.

2. Degree of a polynomial - the degree of the term of highest degree.

7

Examples: LSCC, S. Nunamaker

a. 9 12 93 2 4ab a b- +

b. 7 5 3 64 3 3 2 2x y x y x y- + +

III. Expressions That Are Not Polynomials

a. 2 552x xx

- + b. 20 - x c. y

y

+

+

1

73

IV. Addition and Subtraction of Polynomials

Like terms - terms/expressions that have same variables

raised to the same powers.

Combine/collect like terms

Examples:

a. ( ) ( )- + - + - +5 3 12 7 33 2 3 2x x x x x

b. ( ) ( )8 9 6 32 3 2 3x y xy x y xy- - -

c. ( ) ( )3 2 2 5 8 42 3 2 3x x x x x x- - + - - - +

8

V. Multiplication of Polynomials LSCC, S. Nunamaker

A. (binomial)(binomial) : binomial multiplied by binomial,

use FOIL

( )( )x x+ + =4 3 ( ) ( ) ( ) ( )x x x x× + × + × + ×3 4 4 3

F O I L

= + + +x x x2 3 4 12 = x x2 7 12+ +

Examples:

a. ( )( )x x+ - =5 3

b. ( )( )2 3 5a a+ + =

c. ( )( )x y x y2 23 5+ - =

d. ( )4 1 2x + = ( )( )4 1 4 1x x+ + =

e. ( ) ( )( )5 1 5 1 5 12x x x+ = + + =

f. ( ) ( )( )3 2 3 2 3 22y y y- = - - =

g. ( ) ( )( )2 1 2 1 2 12a a a- = - - =

h. ( )( )2 1 2 1a a- + =

i. ( )( )3 2 3 2y y- + =

9

B. Special Products of Binomials LSCC, S. Nunamaker

1. ( )A B A AB B+ = + +2 2 22

2. ( )A B A AB B- = - +2 2 22

3. ( )( )A B A B A B- + = -2 2

C. Multiplying Two Polynomials

Examples:

a. ( )( )a b a ab b- - + =2 33 2

b. ( )( )4 7 3 2 34 2 2x y x y y y x y- + - =

c. ( )( )2 3 4 2x y x y+ + + =

10

R.4 Factoring LSCC, S. Nunamaker

I. Factoring Terms with Common Factors

Examples:

1. 16 12 4 2+ -x x = 4 4 3 2( )+ -x x

2. 14 352 2 3x y x y- =

3. 12 243 2 2x y x y- =

II. Factoring by Grouping

Pairs of terms have a common factor that can be removed in

a process called factoring by grouping.

*Hint: usually for 4 terms

Examples:

1. x x x3 23 5 15+ - - = x x x2 3 5 3( ) ( )+ - + =( )( )x x+ -3 52

2. p p p3 22 9 18- - + =

3. 4 12 10 304 2 2x x xa a a+ + + =

4. x ax bx ab2 + + + =

III. Factoring Trinomials

Some trinomials can be factored into the product of two

binomials.

A. To factor a trinomial of the form x bx c2 + + , we look for two numbers with a product of

c and a sum of b.

11


1. z z2 2 24- - = ( )( )z z- +6 4

2. x x2 20- - =

3. x x2 18 81- + =

4. y y2 4 21- - =

5. x y xy2 2 18 64- + =

6. x xy y2 25 6+ + =

7. 10 3 2- -x x =

(hint: factoring -1 from the trinomial)

8. a ab b2 28 33+ - =

B. Trinomial of the ax bx x2 + + form can also be factored by grouping. To factor

ax bx c2 + + , first find two factors a c× whose sum is b. Then use factoring by

grouping to write the factorization of the trinomial.

Examples:

a. Factor: 3 11 82x x+ +

Find two positive factors of 24 + Factors of 24 Sum

(ac = ×3 8) whose sum is 11, the 1, 24 25

coefficient of x. 2, 12 14

3, 8 11

The required sum has been found. The remaining factor need

not be checked.

Use the factors of 24 3 11 8 3 3 8 82 2x x x x x+ + = + + +

12

whose sum is 11 to write = + + +( ) ( )3 3 8 82x x x LSCC, S. Nunamaker

11x as 3 8x x+ . = 3 1 8 1x x x( ) ( )+ + +

Factor by grouping. = + +( )( )x x1 3 8

Check: ( )( )x x x x x x x+ + = + + + = + +1 3 8 3 8 3 8 3 11 82 2

b. Factor: 4 17 212z z- -

Find two factors of -84 [ac = × -4 21( )] whose sum is - 17, the

coefficient of z.. Factors of -84 Sum

1, - 84 - 83

- 1, 84 83

2, - 42 - 40

- 2, 42 40

3, - 28 - 25

-- 3, 28 25

4, - 21 - 17

(Once the required sum is found, the remaining factors need

not be checked).

Use the factors of - 84 whose sum is -a7 to write - 17z as

4 21z z- . Factor by grouping. Recall that - - = - +21 21 21 21z z( )

4 17 21 4 4 21 212 2z z z z z- - = + - -

= + - +( ) ( )4 4 21 212z z z

= + - +4 1 21 1z z z( ) ( )

= + -( )( )z z1 4 21

13


1. Factor: 6 11 102x x+ -

2. Factor: 12 32 52x x- +

3. Factor: 30 2 4 2y xy x y+ -

4. Factor: 4 15 42x x+ -

5. Factor: y y2 18 72- +

6. Factor: 5 4 2+ -x x

7. Factor: 4 52a a- -

8. Factor: y y y5 38 15- +

C. Special Factorizations

I. Factors of the Difference of Two Perfect Squares

A B A B A B2 2- = + -( )( )

Examples:

1. x x x2 25 5 5- = + -( )( )

2. 9 252x - =

14

3. 4 812 2x y- = LSCC, S. Nunamaker

4. 25 12x - =

5. x y2 436- =

II. Factors of a Perfect Square Trinomial

A AB B A B2 2 22+ + = +( )

A AB B A B2 2 22- + = -( )

Examples:

1. x x2 8 16+ + =

2. 9 12 42x x+ + =

3. 4 20 252x x- + =

III. To Factor the sum or the difference of two cubes

A B A B A AB B3 3 2 2+ = + - +( )( )

A B A B A AB B3 3 2 2- = - + +( )( )

Examples:

1. a y a y a y a ay y3 3 3 3 2 264 4 4 4 16+ = + = + - +( ) ( )( )

2. 64 1254y y- =

3. 8 3 3 3x y z+ =

4. x y3 3 1- =

5. ( )x y x+ - =3 3

15

LSCC, S. Nunamaker

R.5 Rational Expressions

A rational expression is the quotient of two polynomials.

For example, 3

5

2

3

3

4 5

2

2, ,x

y

y x-

-

- - are rational expressions

I. Domain of a Rational Expression

Any number that makes the denominator zero is not in

the domain of a rational expression.

Examples: Find the domain of the following

1. 5

5x - 2.

x

x x

+

- -

4

4 52 3.

3

42t -

*For a, use set-builder notation : {x x is a real number and x ¹ 5}

4. 3 5

5 62

-

+ +

x

x x 5.

x

x

+

+

1

12 6.

5

3 9

x

x +

II. To simplify rational expressions, use the fact that:

a c

b c

a

b

c

c

a

b

a

b

×

×= × = × =1

Examples: Simplify

1. 6 9

12 18

3 2

2

x x

x x

-

- =

16

LSCC, S. Nunamaker

2. 6 24

12 48

4 3

3 2

x x

x x

-

- =

3. 12 6

6

3 2 3 3

2 2

x y x y

x y

+ =

4. 20 15

15 5 20

2

3 2

x x

x x x

-

- - =

*The simplified version may also be considered as the

equivalent expression of the original expression.

ex. 2

62

-

+ -

x

x x and

-

+

1

3x are equivalent expressions.

III.Multiply or divide and simplify each of the following.

Examples:

1. x

x

x

x x

+

-×

-

- -

4

3

9

2

2

2 =

2. 2 6

3 6

6 12

8 12

2

3 2

x x

x

x

x x

-

-×

-

- =

3. 6

35

12

7

2 4

2 5

3 3

4 5

x y

a b

x y

a b¸ =

4. 4 4

6

3 3

2 2

2 2

2 2

2

2 2

x y

x y

x xy

x y xy

-¸

+

- =

17

LSCC, S. Nunamaker

IV. Adding and Subtracting Rational Expressions

When adding or subtracting rational expressions, it is often

necessary to express the rational expressions in terms of a

common denominator. This common denominator is the

least common multiple (LCM) of the denominators.

Examples:

1. 2 3

2

1

2 62 2

a

a a

a

a a

-

-+

+

- - =

The LCM of ( )a a2 2- and ( )2 62a a- - :

a a a a2 2 2- = -( ); 2 6 2 3 22a a a a- - = + -( )( )

LCM is: a a a( )( )- +2 2 3

2 3

2

2 3

2

2 3

2 3

4 9

2 2 32

2a

a a

a

a a

a

a

a

a a a

-

-=

-

-×

+

+=

-

- +( ) ( )( )

a

a a

a

a a

a

a

a a

a a a

+

- -=

+

- -× =

+

- +

1

2 6

1

2 6 2 2 32 2

2

( )( )

Now they have the LCM, add:

4 9

2 2 3 2 2 3

2 2a

a a a

a a

a a a

-

- ++

+

- +( )( ) ( )( )=

5 9

2 2 3

2a a

a a a

+ -

- +( )( )

2. 3

2 11 15

2

32 2

x

x x

x

x x- ++

-

- =

3. x

x x x x2 211 30

5

9 20+ +-

+ +=

18

LSCC, S. Nunamaker

4. y

y y y2 20

2

4- --

+=

5. x

x y

y

y x2 3 3 2--

-=

6. a

a a

a

a

-

-+

-

-=

3

5

9

252 2

7. 5 5

5 62

x

x

x-

-+ =

V. Complex Rational Expressions

A. Definition

A complex rational expression has rational expressions

in its numerator or denominator or both.

B. Simplifying complex rational expressions

Method 1: Multiply the numerator and denominator by

the LCD of all of the denominators.

Method 2: Take care of the operations in the numerator

and the denominator, then the numerator

multiply by the reciprocal of the

denominator.

19

LSCC, S. Nunamaker

Examples:

1. Method 1:

1 1

1 1

1 1

1 1x y

x y

x y

x y

xy

xy

+

-

=

+

-

× =

Method 2:

1 1

1 1x y

x y

y x

xyy x

xy

+

-

=

+

-=

y x

xy

xy

y x

+´

-=

2.

1 1

1 13 3

a b

a b

+

+

=

3. 5

21

2+

=

4.

25

5a

a

a

-

+ =

20

LSCC, S. Nunamaker

R.6. Radical Notation and Rational Expressions

I. Radical Notation

A number c is said to be an nth root of a if c an = . Square root: if n = 2 Cube root: if n = 3

If given an , the symbol n is called a radical, n is the index, and a is the radicand. The

positive root is called the principal root. To denote a negative root, use - 4, - 8, etc.

Examples: 36 6= , - = -36 6, 32

243

2

35 5= ( ) ,

- = -8 23 , -164 is not a real number

* When a is negative and n is even, an is not a real number.

II. Simplifying Radical Expressions

If a and b are real numbers or expressions for which the

given roots exist. m n, are natural numbers. Here are

some properties of radicals.

A. If n is even, a ann =

B. If n is odd, a ann =

C. a b abn n n× =

D. a

b

a

bn

n

n= ( b ¹ 0 )

E. a amn n m= ( )

Examples: Simplify

1. 325 = 2. 128 2 4c d =

21

3. m n12 24

6

64= 4.

40

5

3

3

m

m= LSCC, S. Nunamaker

5. 27 93 x = 6. -9 4 8a b =

7. 243 5 104 m n = 8. 162 4 6c d =

9. x x2 4 4- + = 10. x x2 16 64+ + =

III. Rationalizing Denominators or Numerators

Removing the radicals in a denominator or numerator.

It is done by multiplying by 1 in such a way so to obtain

a perfect nth power.

Examples:

1. rationalizing the denominator

3

2

3

2

2

2

6

4

6

4

6

2= × = = =

2. rationalizing the numerator

x y x y x y

x y

x y

x y

x y

x y

-=

-×

+

+=

-

+=

-

+5 5 5 5 5 5

2 2( ) ( )

3. Rationalizing the numerator

x y-

3 =

4. Rationalizing the denominator

7

9

3

3=

22

5. Rationalizing the numerator LSCC, S. Nunamaker

50

3 =


2

53 =


8 6

5 2

-

-=


16

93 =


2

3 1-=


3

v w+=

IV. Rational Exponents

For any real number a and any natural numbers m and n,

a an n1/ = a am n mn/ = aa a

m n

m n mn

- = =/

/

1 1

23

Examples:Convert exponent expressions to radical notations LSCC, S. Nunamaker

a. 7 73 4 34/ = b. 8 5 3- / = c a3 5/ =

Examples: Convert to exponential notation and simplify.

1. ( ) ( ) /7 74 5 5 4xy xy= 2. x 36 = 3. 73 =

4. ( )134 5 = 5. 3225 = 6. 743 =

R.7 The Basics of Equation Solving

I. Definitions

A. An equation is a statement that two expressions are equal.

B. Solving an equation in one variable is to find all the values of

the variable that makes the equation true.

C. Solution set is the set of all solutions of an equation.

D. Equivalent equations are equations that have the same solution set.

E. Linear equation in one variable: ax b+ = 0, a and b are real numbers and a ¹ 0.

F. Quadratic equation: ax bx c2 0+ + = , a b c, , are real numbers and a ¹ 0.

G. Empty set: if equation has no solution, denoted by Æ

II. Equation-Solving Principles

A. If a b= , then ac bc= B. If a b= , then a c b c+ = + C. If ab = 0, then a = 0 or

D. If x k2 = , then x k= or x k= - E. If a b= , then a bn n= , for any positive integer n

24

Examples: Solve for the unknown LSCC, S. Nunamaker

a. 3 7 2 14 8 1( ) ( )- = - -x x ------>21 6 14 8 8- = - +x x

21 6 22 8- = -x x

21 6 8 22 8 8- + = - +x x x x

21 2 22+ =x

21 21 2 22 21- + = -x

2 1x =

2 2 1 2x ¸ = ¸

x =1

2

b. 5 5 13- = -x x

c. 7 3 6 11 2( ) ( )x x+ = - +

d. 2 62x x=

e. 5 7+ + =x x

f. x x- + + =3 5 4

25

g. 10 16 6 02x x- + = LSCC, S. Nunamaker

h. 3

2

2 4 4

42m m

m

m++ =

-

-

i. 2 5 3 1y y- - - =

j. n n2 4 4 0+ + =

k. 6 18 02z - =

l. 5 75 02x - =

m. x x2 3 28 0+ - =

n. 3 2 5 7 4 9( ) ( )n n- - = -

26

chapter r-basic concepts of algebra r.1 the real number …mathlady.org/mathlady/mac1105 college...

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