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Exponential Functions
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Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Fractions
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Polynomials
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Fractions
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50


 
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PROPERTIES
English Words Math Symbols
⇒ COMMUTATIVE

(2 - moved)

2 + 3 = 3 + 2

2 · 3 = 3 · 2

⇒ ASSOCIATIVE

(3 - grouped)

(2 + 3) + 5 = 2 + (3 + 5)

(2 · 3) · 5 = 2 · (3 · 5)

⇒ DISTRIBUTIVE

(multiplication over sum)

2 (3 + 5) = 2 · 3 + 2 · 5
⇒ IDENTITY

(result = original)

2 + (0) = 2

3 · (1) = 3

⇒ INVERSE

(result = identity element)

2 + (- 2) = 0

 

Rule of Thumb for signed numbers:

Remember that the sign goes with the number that follows it.

 

Simplifying Expressions

Definition: Two or more terms with the same variable part are called Similar (or Liike) Terms.

Example 1:

Similar (or Liike) Terms:

Example 2:

Simplify each expression (Combine Liike Terms):

a) 7x2 - 2x2

7x2 - 2x2 = (7 - 2)x2 Distributive property
  = 5x2 Addition

b) 5y + 3x – 10y + 8x

5y + 3x – 10y + 8x = 5y – 10y + 3x + 8x Commutative property
  = (5y – 10y) + (3x + 8x) Associative property
  = (5 – 10)y + (3 + 8)x Distributive property
  = - 5y + 11x Addition:

Note: These terms cannot be combined because x ≠ y and the terms are not like or not similar.

Since there are fewer terms in the result than the beginning expression the result is simpler.

 

Example:

Apply the associative property to rewrite each of the following expressions, and then simplify when possible.

1. (5x + 3) + 4 = 5x + (3 + 4) Associative Property
  = 5x + 7 Addition
 
2. 7+ (4 + y) = (7 + 4) + y Associative Property
  = 11 + y Addition
 

The most often missed property is the distributive property. When there is a number or variable multiplied times an expression in parentheses, you must always make sure to multiply times each term inside the parentheses and stop with the "close paren".

Remember that the sign goes with the number that follows it.