Algebra Class: Basic Definitions and Types

Study Notes

Basic Definitions

An algebraic expression is a mathematical phrase that contains variables, constants, and algebraic operations (addition, subtraction, multiplication, division, exponentiation).
Variables represent unknown values and are typically letters (e.g., x, y, z).
Constants are fixed numerical values (e.g., 2, 5, -7).
Algebraic operations are used to combine and manipulate variables and constants. They follow the standard arithmetic rules.

Types of Algebraic Expressions

Monomials: Expressions with only one term (e.g., 3x, -4y², 7).
Binomials: Expressions with two terms (e.g., 2x + 5, x² - 7y).
Trinomials: Expressions with three terms (e.g., 3x² + 2x - 1).
Polynomials: Expressions with one or more terms (e.g., 4x³ - 2x² + x - 5).

Operations on Algebraic Expressions

Addition and Subtraction:
Combine like terms by performing the arithmetic operation on their coefficients.
Like terms have the same variable(s) raised to the same power(s).
Example: 3x + 5x = 8x; 2x² - x² = x²
Example: (2x + 3) + (4x - 1) = 6x + 2
Example: (5x² - 2x + 1) - (2x² + 3x - 4) = 3x² - 5x + 5
Multiplication:
Distribute the multiplier to each term within the expression using the distributive property.
Example: 2x(x + 3) = 2x² + 6x
Example: (x + 2)(x - 3) = x² - x - 6 (using FOIL method, First, Outer, Inner, Last)
Division:
Divide each term of the dividend by the divisor
Example: (6x² + 12x) / 2x = 3x + 6
Example: If dividing by a binomial or higher order polynomial, long division may be needed.

Simplifying Algebraic Expressions

Combine like terms to reduce an expression to its simplest form.
Apply the order of operations (PEMDAS/BODMAS) when necessary.

Evaluating Algebraic Expressions

Replace variables in the expression with numerical values.
Compute the resulting numerical value using standard arithmetic procedures.

Solving Equations with Algebraic Expressions

Use the properties of equality to isolate the variable.
Perform the same algebraic operation on both sides of the equation to maintain equality.

Further Examples

Example of an expression: 3x + 7 - x²
This is a polynomial, in this particular case a trinomial.
Example of a numerical problem: Solve 2x + 5 = 11
Answer: x = 3 (by applying the process of isolation of the variable)

Special Products

Difference of squares: (x - a)(x + a) = x² - a²
Perfect square trinomials: (x ± a)² = x² ± 2ax + a²

Factoring Algebraic Expressions

Factoring is the reverse process of expanding an expression.
It involves identifying common factors and applying the various factorization techniques.
Example: Factoring a quadratic expression like x² + 3x + 2
Answer: (x + 1)(x + 2)

Description

Test your knowledge of algebraic expressions with this quiz focused on definitions and types such as monomials, binomials, trinomials, and polynomials. Learn how to perform operations on these expressions effectively. Suitable for students in Algebra class.