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Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
Solving Linear Equations
Solving Inequalities
Solving Compound Inequalities
Solving Systems of Equations Using Substitution
Simplifying Fractions 3
Factoring quadratics
Special Products
Writing Fractions as Percents
Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
Solving Systems of Equations -
Exponential Functions
Integer Exponents
Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
Decimal Numbers
Negative Numbers
Factoring
Subtracting Polynomials
Adding and Subtracting Fractions
Powers of i
Multiplying and Dividing Fractions
Simplifying Complex Fractions
Finding the Coordinates of a Point
Fractions and Decimals
Rational Expressions
Solving Equations by Factoring
Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
Standard Form for the Equation of a Line
Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
Raising a Product to a Power
Solving Equations with Log Terms on Each Side
Monomial Factors
Solving Inequalities with Fractions and Parentheses
Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
Solving Absolute Value Inequalities
Equations of Circles
Percents and Decimals
Integral Exponents
Linear Equations - Positive and Negative Slopes
Multiplying Radicals
Factoring Special Quadratic Polynomials
Simplifying Rational Expressions
Adding and Subtracting Unlike Fractions
Graphuing Linear Inequalities
Linear Functions
Solving Quadratic Equations by Using the Quadratic Formula
Adding and Subtracting Polynomials
Adding and Subtracting Functions
Basic Algebraic Operations and Simplification
Simplifying Complex Fractions
Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
Graphing Systems of Equations
Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
Solving Systems of Linear Inequalities
Adding and Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Fractions
Solving Linear Equations
Simple Trinomials as Products of Binomials
Solving Nonlinear Equations by Factoring
Solving System of Equations
Exponential Functions
Computing the Area of Circles
The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
Squaring a Difference
Changing the Sign of an Exponent
Adding Fractions
Powers of Radical Expressions
Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Algebra
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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Basic Algebraic Operations and Simplification

We can say that the “basic” algebraic operations you must master are:

  • addition (of two or more terms or expressions)
  • subtraction (of one term or expression from another)
  • multiplication (of one term or expression by another) In the case of multiplication, it turns out to be useful not only to start with two expressions and be able to write down the result of multiplying them together, but also, to be able to start with an expression and rewrite it as the product of two or more factors.
  • division (of one term or expression by another – this leads to the whole subject of working with fractions containing literal symbols.)
  • manipulating radicals or roots, particularly square roots, of algebraic expressions

Because the presence of literal symbols in algebraic expressions can lead to considerable (indeed terrifying) complexity when some of these basic operations are performed, a very important algebraic skill is the ability to simplify algebraic expressions of various sorts, whenever such a thing is possible. Simplification is something you do a lot in algebra (and in mathematics in general), though it is a bit difficult to define precisely what is meant by one expression being simpler than another in all situations. Also, exactly how one might achieve a simplification depends on the features of the algebraic expression with which you are dealing. In general:

one expression is simpler than another if it has fewer terms, or if its parts have fewer terms (for example, in the case of fractions)

The catch is that in the process of simplifying an expression, we must make sure that the new simpler expression is mathematically equivalent to the original expression – it evaluates to the same value as the original expression whenever the same values are substituted for corresponding literal symbols in the two. The rules and strategies for simplification that we will describe are intended to ensure that this requirement is satisfied. By imposing this requirement, we are able to discard the original more complicated expressions and continue to work with the simplified version since in the end the simplified version must give exactly the same results.

When doing algebra, it is generally expected that where an “obvious” simplification of an expression is possible, you will carry out that simplification before stating a final solution to a problem. What are “ obvious ” possible simplifications to check for depends on the situation. The most important and common strategies for simplification of various types of expressions will be described with examples in the next few sections of these notes.

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