Advanced Algebra Flashcards

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.

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.

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.

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 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.

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) ).

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.

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)} ).

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.