Algebraic Expressions and Sets of Numbers

Definition and Structure of Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Understanding their structure is fundamental in algebra.

• Algebraic Expression: A combination of numbers, variables, and operations (addition, subtraction, multiplication, division).

• Example: is an algebraic expression.

• Terms: Parts of an expression separated by addition or subtraction.

• Coefficient: The numerical factor of a term (e.g., 3 in ).

• Constant: A term without a variable (e.g., -5).

Sets of Numbers

Numbers in algebra are classified into different sets, each with unique properties.

• Natural Numbers:

• Whole Numbers:

• Integers:

• Rational Numbers: Numbers that can be written as , where and are integers and .

• Irrational Numbers: Numbers that cannot be written as a fraction (e.g., , ).

• Real Numbers: All rational and irrational numbers.

Polynomials

Definition and Classification

Polynomials are a specific type of algebraic expression with important properties and classifications.

• Polynomial: An expression of the form , where coefficients are real numbers and is a non-negative integer.

• Degree: The highest power of the variable in the polynomial.

• Types:

  • Monomial: One term (e.g., )

  • Binomial: Two terms (e.g., )

  • Trinomial: Three terms (e.g., )

Operations with Polynomials

Polynomials can be added, subtracted, multiplied, and factored.

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or FOIL for binomials.

• Example:

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Common Methods:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials

  • Factoring by grouping

  • Factoring special products (e.g., difference of squares: )

• Example:

Rational Expressions

Definition and Simplification

Rational expressions are quotients of polynomials and are simplified by factoring and reducing common factors.

• Rational Expression: An expression of the form , where and are polynomials and .

• Simplification: Factor numerator and denominator, then reduce common factors.

• Example: (for )

Problem Solving in Algebra

Introduction to Problem Solving

Algebraic problem solving involves translating real-world situations into mathematical expressions and equations.

• Steps:

  1. Read and understand the problem.

  2. Identify what is being asked.

  3. Translate the problem into an equation.

  4. Solve the equation.

  5. Check the solution in the context of the problem.

• Example: If a number plus 5 equals 12, what is the number? Let be the number:

Factoring Polynomials

Methods and Applications

Factoring is used to simplify expressions and solve equations, especially quadratic equations.

• Factoring Trinomials: can often be factored as where and are numbers such that and .

• Example:

• Factoring by Grouping: Used when a polynomial has four terms.

• Example:

Summary Table: Sets of Numbers

The following table summarizes the main sets of numbers used in algebra:

Additional info:

Some referenced videos and sections (e.g., "Central Limit Theorem", "Sampling Distributions") are statistics topics and not directly relevant to Intermediate Algebra. The notes above focus on algebraic expressions, polynomials, rational expressions, and problem solving, which are core to Intermediate Algebra.