Advanced Algebra Flashcards

Number Types

• Natural Numbers: Counting numbers starting from 1 (e.g., 1, 2, 3, 4).
• Whole Numbers: Natural numbers including zero (0, 1, 2, 3, 4).
• Integers: All whole numbers including negative integers and zero.
• Rational Numbers: Can be expressed as a fraction of integers with a non-zero denominator; may result in terminating or repeating decimals.
• Irrational Numbers: Cannot be expressed as fractions; their decimal forms neither terminate nor repeat, such as square roots of non-perfect squares.

Algebraic Properties

• Properties of Natural Numbers: Fundamental rules governing operations on natural numbers.
• Properties for Simplifying Algebraic Expressions: Rules that assist in reducing expressions to simpler forms.
• Properties of Equality: Principles that maintain equality when operations are applied equally to both sides.
• Properties of Inequality: Rules governing inequality expressions.

Quadratics

• Quadratic Formula: Use when a quadratic ( Ax^2 \pm Bx \pm C ) is challenging to factor.
• Discriminant: To determine if a quadratic is factorable, check if ( B^2 - 4AC ) is a perfect square.

Key Concepts

• Absolute Value: Represents the distance of a number from zero, disregarding its sign.
• Additive Inverse: The number that, when added to a given number, results in zero.
• Algebraic Expression: Composed of numbers, variables, and operations.
• Coefficient: A constant multiplying a variable in an expression (e.g., in ( 4xy^2 ), 4 is the coefficient).
• Evaluate: To find the numerical value of an expression.

Multiplicative Concepts

• Multiplicative Inverse: A number that, when multiplied by its reciprocal, equals one.
• Opposite: Two integers that are the same distance from zero, one positive and one negative.
• Reciprocal: Functionally related numbers whose product is one.

Graphing Lines

• X-Intercept: Where the line crosses the x-axis, calculated as ( (C/A, 0) ).
• Y-Intercept: Where the line crosses the y-axis, calculated as ( (0, C/B) ).
• Slope: Defined as the change in y over the change in x, represented as ( (Y2-Y1)/(X2-X1) ).

Line Equations

• Slope-Intercept Form: ( y = mx + b ), where m is the slope and b is the y-intercept.
• Standard Form: Expressed as ( Ax + By = C ).
• Point-Slope Form: Used for a point ( (x_1, y_1) ) and slope m, written as ( y - y_1 = m(x - x_1) ).
• Zero Product Property: If the product of two factors equals zero, at least one factor must be zero.

Algebraic Operations

• F.O.I.L. Method: Used for multiplying two binomials.
• Factorization Techniques: Various rules like the Difference of Squares and products of powers for simplifying expressions.
• Quotient of Powers Rule: ( X^a / X^b = X^{(a-b)} ).
• Power of a Power Rule: ( (X^A)^B = X^{(AB)} ).

Slope Characteristics

• The slope of a vertical line is undefined.
• The slope of a horizontal line is zero.
• Lines that are parallel have identical slopes.
• Lines that are perpendicular have slopes that are negative reciprocals of each other.

Quadratics Summary

• Discriminant analysis ( B^2 - 4AC ):
  • Greater than 0: Two real solutions exist.
  • Equal to 0: One real solution exists.