Real Numbers and Polynomials

Definition: Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers.
Types of Real Numbers:
- Rational Numbers: Can be expressed as a fraction (e.g., 1/2, -3).
- Irrational Numbers: Cannot be expressed as a simple fraction (e.g., √2, π).
Properties:
- Closure: The sum or product of two real numbers is also a real number.
- Associative Property: (a + b) + c = a + (b + c); (ab)c = a(bc).
- Commutative Property: a + b = b + a; ab = ba.
- Distributive Property: a(b + c) = ab + ac.
Number Line: Visual representation of real numbers; positive to the right, negative to the left.

Definition: An algebraic expression consisting of variables raised to non-negative integer powers and coefficients.
Standard Form: Written as ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ) where ( a_n \neq 0 ).
Types:
- Monomial: A single term (e.g., 4x^3).
- Binomial: Two terms (e.g., x^2 + 3).
- Trinomial: Three terms (e.g., x^2 + 2x + 1).
Degree: The highest power of the variable. Example: Degree of ( 3x^4 + 2x^3 ) is 4.
Operations:
- Addition: Combine like terms.
- Subtraction: Combine like terms with negative signs.
- Multiplication: Use the distributive property or FOIL for binomials.
- Division: Polynomial long division or synthetic division.

Definition: Study of geometry using a coordinate system.
Coordinate Plane: Divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
Points: Represented as ordered pairs (x, y).
Distance Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
Midpoint Formula: ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
Slope of a Line: ( m = \frac{y_2 - y_1}{x_2 - x_1} ).
Equation of a Line:
- Slope-Intercept Form: ( y = mx + b ), where m is the slope and b is the y-intercept.
- Point-Slope Form: ( y - y_1 = m(x - x_1) ).
Types of Lines:
- Vertical Lines: Undefined slope, equation x = k.
- Horizontal Lines: Zero slope, equation y = k.

Real numbers encompass all numbers found on the number line, including both rational and irrational categories.
Rational Numbers can be expressed as fractions, such as 1/2 or -3.
Irrational Numbers cannot be represented as simple fractions, examples include √2 and π.
Closure Property ensures that the sum or product of any two real numbers results in another real number.
Associative Property indicates that grouping of numbers does not affect sums or products: (a + b) + c = a + (b + c) and (ab)c = a(bc).
Commutative Property illustrates that the order of addition or multiplication does not impact the result: a + b = b + a and ab = ba.
Distributive Property connects addition and multiplication: a(b + c) = ab + ac.
The Number Line visually represents real numbers with positive values to the right and negative values to the left.

A polynomial is defined as an algebraic expression that consists of variables raised to non-negative integer powers along with coefficients.
Standard Form organizes polynomials as ( a_nx^n + a_{n-1}x^{n-1} +...+ a_1x + a_0 ), where ( a_n ) must not be zero.
Types of Polynomials:
- Monomial: Contains one term, for example, 4x^3.
- Binomial: Composed of two terms, such as x^2 + 3.
- Trinomial: Includes three terms, like x^2 + 2x + 1.
The Degree of a polynomial is determined by the highest power of the variable; for instance, in ( 3x^4 + 2x^3 ), the degree is 4.
Operations:
- Addition involves combining like terms.
- Subtraction also combines like terms, accounting for negative signs.
- Multiplication can be achieved using the distributive property or the FOIL method for binomials.
- Division is performed through polynomial long division or synthetic division.

This field of study focuses on geometry through the use of a coordinate system.
The Coordinate Plane is split into four quadrants by the x-axis (horizontal) and y-axis (vertical).
Points on the plane are represented as ordered pairs in the form (x, y).
The Distance Formula calculates the distance between two points: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
The Midpoint Formula identifies the midpoint between two points: ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
The Slope of a Line is found using the formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} ).
Equations of Lines can be expressed in different forms:
- Slope-Intercept Form: ( y = mx + b ), where m represents the slope and b is the y-intercept.
- Point-Slope Form: ( y - y_1 = m(x - x_1) ).
Types of Lines:
- Vertical Lines have an undefined slope and are expressed as x = k.
- Horizontal Lines possess a zero slope, represented as y = k.