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DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Topic 1: Fundamental Concepts of Algebra
Sub topics:
1
...
2
Exponents
1
...
4
Polynomials
1
...
6
Rational Expressions
-----------------------------------------------------------------------------------------------------------Objectives:
1
...

2
...

3
...

4
...

5
...

6
...

7
...

8
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The objects in a set are called the elements of the set
...

For example: {1, 3, 5, 7, 9}

Important Subsets of the Real Numbers
Name
Natural
numbers, ℕ
Whole
Numbers, 𝕎
Integers

ℤ
Rational
numbers

ℚ

Description
These numbers are used for counting
...


{0, 1, 2, 3, 4, 5, …}

Consist of:
{
...
}

...
, -3, -2, -1}
- zero
{0}
Numbers that can be expressed as a quotient of 2
two integers
...
375

a
8
a and b are integers, b  0 
Q
3
b

 0
...
 0
...

Irrational numbers cannot be expressed as a
quotient of integers
...
414213
...
14159
...

 The lesser of two real numbers is the one farther to the left on a number line
...

Inequalities




less than
less than or equal to

Symbols

ab

Meaning
a is less than or equal to b

ab

b is greater than or equal to a




greater than
greater than or equal to

Example

58
88
85

 4  4

Explanation
Because 5 < 8
Because 8 = 8
Because 8 > 5
Because – 4 = – 4
2

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Absolute Value

 x
x 
 x

x

if x  0
if x  0

represents the distance to the origin from the point x
...


a 0

4
...


a  a

5
...


a a

6
...

Examples of algebraic expressions:

x + 6, x – 6, 6x, x/6, 3x + 5
...
Perform operations within the innermost parentheses and work outward
...

2
...

3
...

4
...

Example: Evaluate the algebraic expression 2
...
5 when
i)
x = 20
ii)
x = 30

3

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Properties of Real Numbers and Algebraic Expressions
Name
Commutative
Property of
Addition
Commutative
Property of
Multiplication
Associative
Property of
Addition

Meaning
Examples
Two real numbers can be added in any  13 + 7 = 7 + 13
order
...

ab = ba
If 3 real numbers are added, it makes  3 + ( 8 + x)
no difference which 2 are added first
...

(a · b) · c = a · (b · c)
Multiplication
distributes
over
addition
...

a+0=a
0+a=a
One can be deleted from a product
...

a + (-a) = 0 and (-a) + a = 0
Inverse Property The product of a nonzero real number
of
and its multiplicative inverse gives 1,
Multiplication
the multiplicative identity
...

Simplify:
1
...
 (5x  13 y  1)

Properties of Negatives
Let a and b represent real numbers, variables, or algebraic expressions
...
(1)a  a

4
...
 (a)  a

5
...
(a)b  ab

6
...
2

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Exponents

bn  b  b  b 
...

Example:
Find a
...


 24

c
...
(4)
...
b n 

1
bn

2
...
b
...
(b m ) n  b mn

bm
4
...
(ab) n  a nb n

n

an
a
7
...
xy 3

2
...
(6 x 4 ) 2

20 x 24
4
...

y



6

3

DIM5058

1
...

a
Radicand

Radical
Sign

Square roots of perfect squares:

a2  a

The Product Rule for Square Roots
Simplified - when it radicand has no factor others than 1 that are perfect squares
...
125x 2

2
...
3x 2

Quotient Rule for Square Roots
If a & b represent non-negative real numbers and

a
a

and
b
b

Examples:

1
...


24 x 4
3x

7

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Adding and Subtracting Square Roots
Examples:

1
...
4 12  2 75

Rationalizing Denominator

2
10

1
...


3
3 7

Other Kinds of Roots
Principal nth Root of a Real Number

a b

 a  bn ;

If n is even,

a  0 , b  0 and

n

If n is odd,

a, b  R
3

Examples: 1
...

3
...

n

an  a

an  a

64 
 32 
4

16 

Perfect nth Power
A number that is the nth power of a rational number is called a perfect nth power
...


3
...


 a
n

m

m

 an

4
...
3 8

2
...
x5

4
4
...


3
...
3 3

9

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Rational Exponents
If

n

represents a real number and n  2 is an integer, then

a

1
n

a  a ;
n

a



1
n

1



a
If

n

a

represent a real number,
m
n

a

m
n




n

1
n
a

m
is a rational number reduced to lowest terms, and n 
n

2 is an integer, the a  ( n a )  a
m
The exponent
consist of two parts:
n
1
...

2
...



1

m

n

m

1
a

m

n

Examples:

1
...
16

5

2

2

3

3
...
4 x

3

4

4
...
4
Polynomials
A polynomial is a single term or the sum of two or more terms containing variables with
whole number exponents
...
g
...

The Degree of axn:
If a  0, the degree of

ax n

is n
...

3x3 + 4x2 –7x +5
Deg
...
3

Deg
...
1

Monomial – one term e
...
7x
Binomial – two terms, each with different exponents
...

Degree of a Polynomial – the highest degree of all the terms of the polynomial
...

Definition of a Polynomial in x
A polynomial in x is an algebraic expression of the form

an x n  a
x n1  a
x n2 
...

The polynomial is of degree n, an is the leading coefficient and a0 is the constant term
...

Example:
1
...
13x  9 x  7 x  1   7 x  2 x  5x  9
3

2

3

2



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DIM5058

MATHEMATICAL TECHNIQUES 1

Multiplying Polynomials
Example: 1
...
(x+5) (x2-5x+25)

Special Products
Let A and B represent real numbers, variables or algebraic expression
...
5
Factoring Polynomial
- the process of writing a polynomial as the product of two or more polynomials
...

Common Factor
Examples: 1
...


x 2 (2 x  5)  17(2 x  5)

2
...


x3  3x 2  4 x  12

Factoring Trinomials
In the forms of ax2 + bx + c
A Strategy for Factoring ax2 + bx + c
(Assume, for the moment, that there is no greatest common factor
...
Find two First terms whose product is ax2:

2
...
By trial and error, perform steps 1 and 2 until the sum of the Outside product and
Inside product is bx:

If no such combinations exist, the polynomial is prime
...
x  13x  40
2

2
...


x 2  144

2
...


36 x 2  49 y 2

Factoring Perfect Square Trinomials
1
...


x  4x  4
2

2
...


A2  2 AB  B 2  ( A  B) 2

4x 2 16x 1

Factoring the Sum and Difference of Two Cubes
1
...


A3  B3  ( A  B)( A2  AB  B 2 )

Examples: 1
...


27 x 3  1

14

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

A Strategy for Factoring a Polynomial
1
...

2
...

c) If there are four or more terms, try factoring by grouping
...
Check to see if any factors with more than one term in the factored polynomial
can be factored further
...

Example: 1
...
x3 – 5x2 – 4x + 20

15

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

1
...

-

Examples: 1
...


x2  2
2x2  4x  6

Excluding Numbers from the Domain
- must exclude numbers from a rational expression’s domain that make the denominator
zero
- Examples: 1
...


x3
x 2  4 x  45

Simplifying Rational Expressions
1
...

2
...


4x  8
Examples: 1
...

x 2  49

Multiplying Rational Expressions
1
...

2
...

3
...


6x  9 x  5

Examples: 1
...


x 2  5x  6 x 2  9

x2  x  6 x2  x  6

16

DIM5058

MATHEMATICAL TECHNIQUES 1

TOPIC 1

Dividing Rational Expressions
Examples: 1
...


x2  x
x2 1

x 2  4 x 2  5x  6

Adding and Subtracting Rational Expressions with Different Denominator

a c ad  bc
 
b d
bd

;

a c ad  bc
 
b d
bd ,

b  0, d  0

Examples:
1
...

3
...


- Examples:

1
...


x 1
x2
3 1
x2  4

17



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