Real Numbers and Their Properties

P.2 Real Numbers and Their Properties

Natural Numbers 1,2,3 ….
Whole Numbers 0, 1, 2, . . .
Integers . . . –2, -1, 0, 1, 2 . . .
Real Numbers
Rational numbers (p/q) Where p & q are integers, q ≠0
Irrational numbers non-terminating and non-repeating)

Properties of Negation
Let a and b be real numbers, variables or algebraic expressions.

1. (−1)a=−a
2. −(−a)=a
3. (−a)b=−(ab)=a(−b)
4. (−a)(−b)=ab
5. −(a+b)=(−a)+(−b)

Properties of Fractions

1. Equivalent fractions: if and only if ad = bc.
2. Rules of signs : and −
3. Generate equivalent fractions:
4. Add or subtract with like denominators :
5. Add or subtract with unlike denominators:
6. Multiply fractions:
7. Divide fractions:

Examples:

P.3 The Real Number Line and Order

Definition of Order on the Real Number Line
a < b and b > a, for a and b real numbers, we say that a is less than b or b is greater
than a.

Sets and Intervals
A = {1, 2, 3, 4, 5, 6, 7} A = {x| x is an integer and 0 < x < 8}
U = Union ∩ = Intersection

Definition of Absolute Value
Let x be a real number. The absolute value of x denoted by |x|, is

Example:

|2x| = 10

Properties of Absolute Values
Let a and b be real numbers. Then the following properties are true.

Distance Between Two Numbers
Let a and b be real numbers. The distance between a and b is given by
Distance = |b – a| = |a – b|

Example: distance between 3 and 6

P.4 Integer Exponents

Properties of Exponents

Property	Example
aman=am+n
(ab)m=ambm
a0 =1

Example:

Scientific Notation

Example:

I nterest Formulas
Simple Interest -- A =P(1 +rt) Compound Interest --

Example:
How much savings would you have if you invested $4000 for 5 years at 4%?
a) simple interest

b) compounded daily

P.5 Radicals and Rational Exponents

In the expression , is called a radical , the number n is called the index, and x
is called the radicand. If no index is given, it is assumed to be 2.

Properties of radicals

Property	Example

Examples:

Simplify the following:

Rationalizing a denominator

Rationalize the numerator

Addition and subtraction of radical expressions

Example:

Convert to solve :

P.6 Algebraic Expressions

A polynomial in x is an ex pression of the form

Sums and Differences of Polynomials

Example:

Example:

Multiplying Binomials – FOIL method or distributive method

or

FOIL (first, outside, inside, last)

Example:

Example:

Special Products

Difference of squares

Square of a Binomial

Cube of a Binomial

Example:
An open box is made by cutting squares from the corners of a piece of metal that
measures 10 by 12 inches and turning up the sides. The sides of the cut-out squares
are all x inches long, so the box is x inches tall. Find the volume when x = 2.

P.7 Factoring

Removing Common Factors

Example:

Factoring Special Polynomial Forms

Difference of squares

Perfect Squares

Difference of cubes

Sum of cubes
 

Factoring the Difference of Squares

Example:

Example:

Factoring Cubes
Example:

Factoring a Trinomial
Example:

Factoring by grouping
Example:

Factoring Guidelines
a) If the polynomial has a greatest common factor other than 1, then factor out the
greatest common factor.

b) If the polynomial has two terms, then see if is the difference of two squares,
or the sum or difference of two cubes. Remember if it is the sum of two squares it
will not factor.

c) If the polynomial has three terms, then it is either a perfect square trinomial or if it
is not in which case you use one of the trial and error methods.

d) If the polynomial has more than three terms, then try to factor it by grouping.

Final check, look and see if any of the factors you have written can be factored further

P.8 Rational Expressions

Domain of an expression- All the x values that are valid .

The domain for all polynomials -

Domain for rational expression –
Find the domain

Domain for radicals –

Find the domain

Simplifying a Rational Expression
Example:
 

Multiplying a Rational Expression
Example:

Dividing a Rational Expression

Example:

Combining Rational Expressions

The LCD of two or more rational expressions is found as follows:

1. Factor each denominator completely .
2. Identify each different prime factor from all the denominators.
3. Form a product using each different factor to the highest power that occurs in any
one denominator. This product is the LCD.
Example:

Example:

Compound Fractions

Example: