代数基本概念与运算

Fundamental Concepts of Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in mathematical expressions and equations.
• Variables are symbols, often letters (like x, y, or z), that represent unknown values.
• Constants are fixed values.
• Expressions combine variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponentiation).
• Equations are statements that show the equality of two expressions.

Basic Operations in Algebra

• Addition: Combining like terms (e.g., 3x + 2x = 5x).
• Subtraction: Subtracting like terms (e.g., 5y - 2y = 3y).
• Multiplication: Expanding expressions, using the distributive property (e.g., 2(x+3) = 2x + 6). Multiplying exponents: x^a * x^b = x^(a+b)
• Division: Simplifying expressions, including dividing polynomials. When dividing exponents, subtract the exponent in the denominator from the exponent in the numerator (e.g., x^5 / x^2 = x^3 ).
• Exponents: Repeated multiplication of a number (e.g., x^3 = x * x * x).

Solving Equations

• Addition Property of Equality: Adding the same quantity to both sides of an equation maintains equality.
• Subtraction Property of Equality: Subtracting the same quantity from both sides of an equation maintains equality.
• Multiplication Property of Equality: Multiplying both sides of an equation by the same quantity maintains equality.
• Division Property of Equality: Dividing both sides of an equation by the same quantity (excluding zero) maintains equality.
• Combining like terms: Simplify an equation by consolidating terms with the same variables and powers.
• Distributive Property: Important for expanding expressions involving parentheses (a(b + c) = ab + ac).
• Solving linear equations: Solving for the unknown variable in an equation that involves only first powers of the variable.

Types of Equations

• Linear Equations: Equations that represent a straight line on a graph; the highest power of the variable is 1 (e.g., 2x + 5 = 11.)
• Quadratic Equations: Equations representing a parabola on a graph; the highest power of the variable is 2 (e.g., ax^2 + bx + c = 0). Methods for solving include factoring, completing the square, and the quadratic formula.
• System of Equations: Multiple equations with multiple variables, solved simultaneously. Solutions may be unique, infinitely many, or not exist.
• Inequalities: Show a relationship where one value is greater than, less than, greater than or equal to, or less than or equal to another value.

Functions

• A function is a relationship between two sets of numbers where each input has one and only one output.
• The input is often represented by 'x' (domain) and the output by 'y' (range or image).
• Function notation: f(x) = ... represents the output of the function f when the input is x.

Polynomials

• Polynomials are algebraic expressions consisting of variables and coefficients, combined using only the operations of addition, subtraction, and multiplication.
• Key concepts include identifying the degree, leading coefficient, and terms of a polynomial.

Factoring Polynomials

• Expressing a polynomial as a product of simpler polynomials.
• Factoring is crucial for simplifying expressions, solving equations, and working with functions. Common factoring techniques include the greatest common factor (GCF), difference of squares, and trinomial factoring.

Graphing

• Graphing equations helps visualize relationships between variables.
• Plotting points and understanding the shape of the graph based on the equation is important. Understanding the x and y intercepts is essential.
• Slope-intercept form of an equation is useful for graphing linear equations.