1
Fundamental Concepts of Algebra
1.1 Real Numbers
Objective: Students will be introduced to the
real number system that is used throughout
mathematics and will be acquainted with the
symbols that represent them.
The Real Numbers
The real numbers can be ordered and
represented in order on a number line
0
-1.87
2
4.55
-3 -2 -1 0 1 2 3 4
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
REAL NUMBERS (R)
Definition:
REAL NUMBERS (R)
- Set of all rational and
irrational numbers.
SUBSETS of R
Definition:
RATIONAL NUMBERS (Q)
- numbers that can be expressed as a
quotient a/b, where a and b are integers.
- terminating or repeating decimals
- Ex: {1/2, .25, 1.3, 5}
SUBSETS of R
Definition:
IRRATIONAL NUMBERS (Q´)
- infinite and non-repeating decimals
- Ex: { ∏, √2, -1.436512…..}
SUBSETS of R
Definition:
INTEGERS (Z)
- numbers that consist of positive
integers, negative integers, and zero,
- {…, -2, -1, 0, 1, 2 ,…}
SUBSETS of R
Definition:
NATURAL NUMBERS (N)
- counting numbers
- positive integers
- {1, 2, 3, 4, ….}
SUBSETS of R
Definition:
WHOLE NUMBERS (W)
- nonnegative integers
- {0, 1, 2, 3, 4, …}
The Set of Real Numbers
Q'
Q
W
Z
Q
N
PROPERTIES of R
Definition:
CLOSURE PROPERTY
Given real numbers a and b,
Then, a + b is a real number (+),
or a x b is a real number (x).
PROPERTIES of R
Example 1:
12 + 3 is a real number. Therefore, the
set of reals is CLOSED with respect to
addition.
PROPERTIES of R
Example 2:
12 x 4.2 is a real number. Therefore, the
set of reals is CLOSED with respect to
multiplication.
PROPERTIES of R
Definition:
COMMUTATIVE PROPERTY
Given real numbers a and b,
Addition: a
+b=b+a
Multiplication: ab = ba
PROPERTIES of R
Example 3:
Addition:
2.3 + 1.2 = 1.2 + 2.3
Multiplication:
(2)(3.5) = (3.5)(2)
PROPERTIES of R
Definition:
ASSOCIATIVE PROPERTY
Given real numbers a, b and c,
Addition:
(a + b) + c = a + (b + c)
Multiplication: (ab)c = a(bc)
PROPERTIES of R
Example 4:
Addition:
(6 + 0.5) + ¼ = 6 + (0.5 + ¼)
Multiplication:
(9 x 3) x 4 = 9 x (3 x 4)
PROPERTIES of R
Definition:
DISTRIBUTIVE PROPERTY of
MULTIPLICATION OVER
ADDITION
Given real numbers a, b and c,
a (b + c) = ab + ac
PROPERTIES of R
Example 5:
4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)
Example 6:
2x (3x – b) = (2x)(3x) + (2x)(-b)
PROPERTIES of R
Definition:
IDENTITY PROPERTY
Given a real number a,
Addition: 0 + a = a
Multiplication: 1 x a = a
PROPERTIES of R
Example 7:
Addition:
0 + (-1.342) = -1.342
Multiplication:
(1)(0.1234) = 0.1234
PROPERTIES of R
Definition:
INVERSE PROPERTY
Given a real number a,
Addition: a
+ (-a) = 0
Multiplication: a x (1/a) = 1
PROPERTIES of R
Example 8:
Addition:
1.342 + (-1.342) = 0
Multiplication:
(0.1234)(1/0.1234) = 1
Inequalities, graphs, and notation
Inequality
Graph
3 x 7
(
]
3
7
x5
Interval
(
3,7
5,
5
1
x
3
1

 ,  
3

]

1
3
) or ( means not included in the solution
] or [ means included in the solution
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Intervals
Interval
Graph
Example
(a, b)
a
b
[a, b]
a
[
b
]
(a, b]
a
(
b
]
[a, b)
a
b
[
)
(a,  )
a
(
(- , b]
(3, 5)
[4, 7]
(-1, 3]
[-2, 0)
(1,  )
(
b
(- , b)
[a,  )
)
)
a
[
b
]
3
5
(
)
4
7
[
]
-1
3
(
]
-2
0
[
)
1
(
2
(- , 2)
)
[0,  )
0
(- , -3]
-3
[
]
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Absolute Value
 a if a  0
a 
a if a  0
To evaluate:
3  8  5  (5)  5
Notice the
opposite sign
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
1-C
Real Number Venn Diagram
Scientific Notation
A short-hand way of writing
large numbers without
writing all of the zeros.
When using Scientific Notation,
there are two kinds of exponents:
positive and negative
Positive Exponent:
2.35 x 108
Negative Exponent:
3.97 x 10-7
An easy way to remember this is:
• If an exponent is positive, the number gets
larger, so move the decimal to the right.
• If an exponent is negative, the number
gets smaller, so move the decimal to the
left.
The exponent also tells how many
spaces to move the decimal:
4.08 x 103 = 4 0 8
In this problem, the exponent is +3, so
the decimal moves 3 spaces to the right.
The exponent also tells how many
spaces to move the decimal:
4.08 x 10-3 =
408
In this problem, the exponent is -3, so the
decimal moves 3 spaces to the left.
Try changing these numbers from
Scientific Notation to Standard
Notation:
1) 9.678 x 104
96780
2) 7.4521 x 10-3
.0074521
3) 8.513904567 x 107
85139045.67
4) 4.09748 x 10-5
.0000409748
When changing from Standard
Notation to Scientific Notation:
1) First, move the decimal after the first whole number:
3258
2) Second, add your multiplication sign and
your base (10).
3 . 2 5 8 x 10
3) Count how many spaces the decimal moved
and this is the exponent.
3 . 2 5 8 x 10 3
3 2 1
When changing from Standard
Notation to Scientific Notation:
4) See if the original number is greater than or less
than one.
– If the number is greater than one, the exponent
will be positive.
348943 = 3.489 x 105
– If the number is less than one, the exponent will
be negative.
.0000000672 = 6.72 x 10-8
Try changing these numbers from
Standard Notation to Scientific
Notation:
1) 9872432
9.872432 x 106
2) .0000345
3.45 x 10-5
3) .08376
8.376 x 102
4) 5673
5.673 x 103
1-1 Answers (2-40e, 50,52)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
2. -,-,+,+
4. >,<,=
6. <,>,>
8. b > 0, s < 0, w > -4, 1/5< c < 1/3, p < -2, -m > -2, r/s ≥ 1/5, 1/f ≤ 14, |x| < 4
10. 10, 3, 17
12. 4, 5/2, 10
14. √3 -1.7, √3 – 1.7, 2/15
16. 4,6, 6, 10
18. 12, 3, 3, ,9
20. | -√2-x|> 1
22. |4-x | < 2
24. |x + 2| > 2
26. x – 5
28. 7 + x
30. a – b
32. x2 + 1
34. =
36 ≠
38 ≠
40 ≠
50. 8.52 x 104
5.5 x 10-6
2.49 x 107
52. 23,000,000 .00000000701
12,300,000,000
1.2 Laws of Exponents
Law
Example
mn
x x x
x
mn
3 
3
a a a
m n
a 
m
n
a
m
a
mn

a
n
a
 ab   a b
n
n
n n
n
a
a
 
   n
b
b
312
3 12
5
6
3
5(6)
15
30
y14
14 12
2
y
y
12
y
4 4
4
3
r

3
r

81
r
 
4
3
4
64
 4
   3 3
x
x
 x
3
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Exponents
n,m positive
integers
Definition
n factors
a
n
a
0
a 1
 a  0
n
n
1
 n
a
 a  0
a
a
a
m/ n
m / n
a  a  a  a  ...  a
n
0
a
am / n  n am
a
m / n

1
n
am
Example
5  5  5  5  125
3
32  1
1
1
4
2  4 
16
2
0
1252 / 3  3 1252  25
4
 
9
3 / 2
3
27
9
   
8
4
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
http://www.youtube.com/watch?v=QIZTruxt2
rQ&feature=related
1.2 Answers: p. 29 (12-30 x3)
12. -12x2
18.
12
y5
24. -4x12y7
30. -288r8s11
Definitions
x is read the " square root of x."
The
is called the radical sign.
x
The expression inside the radical sign
is called the radicand.
The entire expression, including the radical
sign and radicand, is called the radical
expression.
Definitions
The positive or principal square root of a
positive number a is written as a . The
negative square root is written as - a .
a  b if b  a
2
Also, the square root of 0 is 0, written 0  0.
Note that the principal square root of a positive number, a,
is the positive number whose square equals a. Whenever
the term ‘square root’ is used in this book, the positive or
principal square root is meant to be used.
Definitions
The index tells the “root” of the expression.
Since square roots have an index of 2, the
index is generally not written in a square root.
x means
2
x
Example:
25  5 (since 5  5  5  25)
2
9 3
 (since
16 4
2
 3   3  3  9
       )
 4   4  4  16
Definitions
Square roots of negative numbers are not
real numbers. Square roots of negative
numbers are called imaginary numbers.
 25  ?
There is no number multiplied by itself
that will give you –25.
(Imaginary numbers will be discussed in a later section)
Cube and Fourth Roots
3
4
3
a is read “the cube root of a.”
a is read “the fourth root of a.”
a  b if b  a
3
4
a  b if b  a
4
3
8  2 since 2  2  2  8
3
 8  2 since (2)( 2)( 2)  8
4
81  3 since 3  3  3  3  3  81
4
Even and Odd Indices
Even Indices
The nth root of a, n a , where n is an even
index and a is a nonnegative real number,
is the nonnegative real number b such that
bn = a.
4
81  3 since 34  81
100  10 since 10  100
2
Even and Odd Indices
Odd Indices
The nth root of a, n a , where n is an odd
index and a is a any real number, is the
real number b such that bn = a.
3
5
64  4 since 43  64
 32  2 since (-2)  32
5
Cube and Fourth Roots
Note that the cube root of a positive
number is a positive number and the cube
root of a negative number is a negative
number.
The radicand of a fourth root (or any even
root) must be a nonnegative number for
the expression to be a real number.
Evaluate by Using Absolute Value
For any real number a,
a a
2
72  7  7
(9) 2   9  9
(9a  11b) 2  9a  11b
( x 2  12 x  36)  ( x  6) 2  ( x  6)
Changing a Radical Expression
A radical expression can be written using exponents
by using the following procedure:
n
a a
1
n
When a is nonnegative, n can be any index.
When a is negative, n must be odd.
7 7
3
9
1
2
 
 3x  7 z 
x y x y
4
3x  7 z
4
4
13
4 19
Changing a Radical Expression
n
a a
1
n
When a is nonnegative, n can be any index.
When a is negative, n must be odd.
Exponential expressions can be converted to
radical expressions by reversing the procedure.
151 2  15
b1 3  3 b
Simplifying Radical Expressions
This rule can be expanded so that radicals of the
form n a m can be written as exponential expressions.
For any nonnegative number a, and integers m and n,
Power
n
8x
a 
m
 a
n
m
 am n
Index
b2 3  3 b2
2
 9 y
73

 8x
3
2
9y

7
Definitions
A perfect square is the square of a natural
number. 1, 4, 9, 16, 25, and 36 are the first six
perfect squares.
Variables with exponents may also be perfect
squares. Examples include x2, (x2)2 and (x3)2.
A perfect cube is the cube of a natural number.
1, 8, 27, 64, 125, and 216 are the first six perfect
cubes.
Variables with exponents may also be perfect
cubes. Examples include x3, (x2)3 and (x3)3.
Perfect Powers
This idea can be expanded to perfect powers of a
variable for any radicand.
The radicand xn is a perfect power when n is a
multiple of the index of the radicand.
A quick way to determine if a radicand xn is a
perfect power for an index is to determine if the
exponent n is divisible by the index of the radical.
Example: 5 x 20
Since the exponent, 20, is divisible by
the index, 5, x20 is a perfect fifth power.
Product Rule for Radicals
For nonnegativ e real numbers a and b,
n
a  b  ab
n
n
Examples:
3
32 
3
 250  3  125 3 2  53 2
4
48  4 16 4 3  24 3
3
8 4 2 4
3
3
Product Rule for Radicals
To Simplify Radicals Using the Product Rule
1. If the radicand contains a coefficient other than 1, write it
as a product of the two numbers, one of which is the largest
perfect power for the index.
2. Write each variable factor as a product of two factors, one
of which is the largest perfect power of the variable for the
index.
3. Use the product rule to write the radical expression as a
product of radicals. Place all the perfect powers under the
same radical.
4. Simplify the radical containing the perfect powers.
Product Rule for Radicals
Examples:
72  36  2  36 2  6 2
4
3
b
23
 b b  b
20 3
4
4
16 x y  8 x y
3
6
 32 x y
4
18
3
31
3
20 4
63
b | b | b
2  2 xy
  16 x y
4
 2 x 4 | y 7 | 4 2 x 2 y 3
3
16
28 4
5
4
23
2
2
2x y
3
3
*When the radical is simplified, the
radicand does not have a variable with
an exponent greater than or equal to
the index.
Quotient Rule for Radicals
For nonnegativ e real numbers a and b,
n
n
a n a

, b0
b
b
Examples:
81

100
81
9

10
100
75
 Simplify radicand, if possible.
3
25

1
25  5
Quotient Rule for Radicals
More Examples:
3
64 x 6

12
y
64 x 5
2x
4
3

64 x 6
3
3
y12
4x2
 4
y
64 x 5

3
2x
32 x 2
16 x 2 2

 4x 2
1
1
8
8
4
3a 6b5
3
a
3
a
4



 2 13
8
8
4
16a b
16b
16b
4
4
a8 4 3
16b8
a2 4 3

2b 2
CAUTION!
The product rule does not apply to
addition or subtraction!
a  b  a b
a  b  ab

Rationalizing Denominators
To Rationalize a Denominator
Multiply both the numerator and the denominator of the fraction
by a radical that will result in the radicand in the denominator
becoming a perfect power.
2
2 3
6
Cannot be simplified further.



3
3
3 3
Examples:
x2

3
y
x2
y
3

y3
y
3

x 2 y 3 xy y | x | y


3
3
y
y
y2
5 pq 4
10 pq 4 r q 2 10 pr
5 pq 4
2r




2r
2r
2r
2r
2r
Conjugates
When the denominator of a rational expression is a
binomial that contains a radical, the denominator is
rationalized. This is done by using the conjugate of
the denominator. The conjugate of a binomial is a
binomial having the same two terms with the sign
of the second term changed.
The conjugate of 5  6 is 5  6
The conjugate of 3x  4 2 y is 3x  4 2 y
Simplifying Radicals
Simplify by rationalizing the denominator:
5
5
2  1 5( 2  1)



2 1
2  1 2 1
2 1
c  2d

c d
c  2d

c d
c d

c d
( c  2d )( c  d ) c  cd  2cd  d 2

cd
( c  d )( c  d )
Simplifying Radicals
A Radical Expression is Simplified When the Following
Are All True
1. No perfect powers are factors of the radicand and all
exponents in the radicand are less than the index.
2. No radicand contains a fraction.
3. No denominator contains a radical.
Assignment:
• Day 2: Continued…pp. 29-31 (3-9, 33-81
x3, 92, 101/102)
• Day 3: Continued…pp. 29-31 (3-9, 33-81
x3, 92, 101/102)
Even Answers:
Day 2: Continued…
pp. 29-31 (3-9, 33-81 x3, 92, 101/102)
4. ½
6.5/1
8.243/1
36.4r5/6
42.-y11/2
48.x5/3
54.a) 4+ x √x b) (4+x) √(4+x)
60.4
66.3 r s2 4√r
72.xy3 /5 • 4√5x2
78. 5x2 y5 √2
92. (a) -1.0813
(b) -44.3624
1.3 Algebraic Expressions
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Polynomials
• Addition
 3x
3
 
 2 x  7 x  15  5 x  13x  12
2
3

3x3  2 x 2  7 x  15  5 x3  13x  12
8 x3  2 x 2  6 x  27
Combine like terms
• Subtraction

 
x3  x 2  6 x  1  3x3  x 2  2 x

x3  x 2  6 x  1  3x3  x 2  2 x Distribute
2 x3  4 x  1 Combine like terms
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Polynomials
• Multiplication
 2 x  53x  2
2 x(3x  2)  5(3x  2) Distribute
6x  4x 15x 10
2
6 x 11x 10
2
Distribute
Combine like terms
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Polynomials
• Division
http://www.youtube.com/watch?v=uERRlYWmmU
1.3 (4-44 x 4) Answers Day 1
4. 6 x 3  10 x 2  2 x
2
2
8. 4 x  23xy  15 y
12. 7 x 4  11x 3  4 x 2  42 x  24
16. 2 x 6  x 5  10 x 4  3x 3  x 2  10 x  5
2
20. 6 xz  y
24. 9 x 2  y 6
28. 25x 2  40 xy  16 y 2
32. x 4  2 x 2 y 2  y 4
36. x  y
40. 27 x3  108x 2 y  144 xy2  64 y 3
44. x 2  4 y 2  9 z 2  4 xy  6 xz  12 yz
1.3 Factoring Polynomials
• Greatest Common Factor
6t  36t
6t 2  t  6 
3
2
The terms have 6t2 in common
• Grouping
mx2  mx  2x  2
Factor mx
Factor –2
mx  x  1  2  x  1
 mx  2 x  1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Factoring Polynomials
• Difference of Two Squares:
x 2  y 2   x  y  x  y 
Ex. m  9
2
 m  3 m  3
• Sum/Difference of Two Cubes:

x3  y 3   x  y  x 2
Ex. 8 x  1
xy  y 2

3
 2 x  1  4 x
2

 2x 1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Factoring Polynomials
• Trinomials
Ex. x  5x  6
2
 x  3 x  2
Trial and Error
Ex. 6 x  27 x  12 x
3

2
3x 2 x  9 x  4
2

Greatest Common Factor
3x  2 x  1 x  4  Trial and Error
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
http://www.youtube.com/watch?v=OFSrINhf
NsQ
POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.
What is the degree and leading
coefficient of 3x5 – 3x + 2 ?
POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are called
QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are called
CUBIC POLYNOMIAL FUNCTIONS
1.3 Answers (48-100 x 4) Day 2
48. 5 xy(2  3 y )
52. 11r 2 s 3 (11rs  7s  5r 2 )
56. Irreducible
60. (3 x  5)(7 x  2)
64. (4 z  7) 2
68. (9r  4t )(9r  4t )
72. x( x  5)( x  5)
76. 4(4 x  3 y )( 4 x  3 y )
80. (6 x3  5 y)(36 x6  30 x3 y  25 y 2 )
84. (ay  3x)( 2 y  x)
88. ( x  2)( x  3)( x 2  2 x  4)
92. ( x 4  4)( x 2  2)( x 2  2)
96. ( y  3  2 x)( y  3  2 x)
100. x(2 x  1) 2
1.4 Rational Expressions
Operation
P, Q, R, and S are polynomials
Addition
P Q PQ
 
R R
R
Subtraction
P Q P Q
 
R R
R
Multiplication
P Q PQ
 
R S RS
Division
Notice the
common
denominator
P Q P S PS
   
R S R Q RQ
Reciprocal
and Multiply
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Rational Expressions
• Simplifying
Factor
Cancel common
factors
x  5  x  5 

x 2  25
x5


2
 x  2  x  5  x  2
x  7 x  10
• Multiplying
Factor
Cancel common
factors
6  x  1
x2  2 x  1 6 x 2  6 x  x  1 x  1 6 x  x  1



 2
3 2
3
x
 x  1 x  1
x2
x
x 1
Multiply Across
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Rational Expressions
• Adding/Subtracting
3
2

x x4
3 x  4
2x


x( x  4)  x  4  x
Must have LCD:
x(x + 4)
3x  12  2 x
5 x  12


x( x  4)
x  x  4
Distribute and
combine fractions
Combine like terms
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Other Algebraic Fractions
• Complex Fractions
3

3

2
2

x
x


x

9
9


 4x

4
x
x


x
x

Multiply by the
LCD: x
3  2x
3  2x
1



2
9  4x
 3  2 x  3  2 x  3  2 x
Distribute and
reduce to get here
Simplify to
get to here
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
Other Algebraic Fractions
Notice:

a b


a  b  a b
• Rationalizing a Denominator

7 3 y
7

3 y
3 y 3


21  7 y

9 y

y

Multiply by
the conjugate
Simplify
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc .
1.4 Answers (3-30 x 3)
6.
2x 1
3x  2
12.
1
x 2 ( x  2)
18.
5s 2  2 s  4
(5s  2) 2
24.
2(3 x  1)
x
30.
u 2  22u  10
u (u  5)
1.4 Answers Day 2 (33-51 x 3)
rs
36. r 2  s 2
42.
2
ax
 10
48.
(2 x  2h  3)( 2 x  3)
Answers to Ch. Review
1.
Positive
15.
2.
84
16.
3.
6-x
17.
4.
5.
6.
7.
8.
3.865 x 102
0.000093
1.76 x 1013
4x2y4
9.
10.
11.
12.
13.
14.
𝟏
𝒙𝟒 𝒚𝟐
𝟐𝟐𝒙𝒚
𝟏𝟏𝒚𝟐
𝒎 𝒎
𝒄
+𝟏
17x3 - 6x + 3
12x3 + 73x2 + 79x – 52
x4 + 13x2 – 14
64x3 + 336x2y + 588xy2 + 343y3
𝒙+𝟖
𝒙+𝟗
𝒂
(𝒂𝟐 +𝟐𝟓)(𝟐𝒂+𝟓)
𝟓(𝟓𝒙+𝟕)
(𝟕𝒙−𝟐𝟓)
Simplifying Radicals Video
http://www.youtube.com/watch?v=pZSuMBX
zEic
Complex Fractions Video
• http://www.wonderhowto.com/how-tosimplify-complex-fractions-algebra-365934/
Negative Exponents Video
http://www.youtube.com/watch?v=c4aiYf3fz
VQ
Rational Expressions Video
http://www.youtube.com/watch?v=L1KDC0lWsY