Number Systems Overview

Definition: A number system is a way of representing numbers using a consistent set of symbols and rules.
Types of Numbers:
- Natural Numbers: Positive integers (1, 2, 3, …).
- Whole Numbers: Natural numbers plus zero (0, 1, 2, …).
- Integers: Whole numbers and their negative counterparts (… -3, -2, -1, 0, 1, 2, 3 …).
- Rational Numbers: Numbers that can be expressed as the quotient of two integers (a/b where b ≠ 0).
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
- Real Numbers: All rational and irrational numbers combined.
Properties:
- Closure Property: For any two numbers in a set, their sum or product is also in the set.
- Commutative Property: Order does not affect the sum or product (a + b = b + a).
- Associative Property: Grouping does not affect the sum or product ((a + b) + c = a + (b + c)).
- Distributive Property: a(b + c) = ab + ac.
Number Line: A visual representation of numbers, showing the order and magnitude.

Definition: An algebraic expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
Standard Form: A polynomial is expressed in descending order of the powers of its variable (e.g., ax^n + bx^(n-1) + ... + k).
Types of Polynomials:
- Monomial: A polynomial with one term (e.g., 4x).
- Binomial: A polynomial with two terms (e.g., 3x^2 + 2).
- Trinomial: A polynomial with three terms (e.g., x^2 + 5x + 6).
- Multinomial: A polynomial with more than three terms.
Degree of a Polynomial: The highest power of the variable in the polynomial (e.g., degree of 4x^3 + 2x^2 + 1 is 3).
Operations with Polynomials:
- Addition and Subtraction: Combine like terms.
- Multiplication: Use distributive property (FOIL for binomials).
- Division: Can be performed using long division or synthetic division.
Factorization: Expressing a polynomial as a product of its factors (e.g., x^2 - 9 = (x - 3)(x + 3)).

Rational Root Theorem: Provides a method to find possible rational roots of a polynomial.
Theorems:
- Fundamental Theorem of Algebra: A polynomial of degree n has exactly n roots, counting multiplicities.
- Remainder Theorem: When a polynomial f(x) is divided by (x - a), the remainder is f(a).
Graphing: The graphical representation of polynomials can show intercepts, turning points, and end behavior based on the degree and leading coefficient.

A number system provides a consistent framework for representing numbers using specific symbols and rules.
Natural Numbers include the positive integers like 1, 2, 3, and so on.
Whole Numbers comprise natural numbers along with zero, forming the set {0, 1, 2, …}.
Integers extend whole numbers to include negatives, covering all positive and negative whole numbers and zero.
Rational Numbers can be expressed as the quotient of two integers (a/b), where the denominator cannot be zero.
Irrational Numbers cannot be represented as a simple fraction and include examples like the square root of 2 (√2) and pi (π).
Real Numbers encompass both rational and irrational numbers, forming the complete set of values on the number line.
Closure Property ensures that the sum or product of any two numbers within a set remains within that set.
Commutative Property states that the order of addition or multiplication does not change the result (e.g., a + b = b + a).
Associative Property indicates that the grouping of numbers does not affect their sum or product ((a + b) + c = a + (b + c)).
Distributive Property combines multiplication with addition (a(b + c) = ab + ac).
A Number Line visually represents numbers, illustrating their order and comparative size.

A polynomial is an algebraic expression formed by variables and coefficients, connected through addition, subtraction, multiplication, and non-negative integer exponents.
Standard Form refers to writing a polynomial in descending order of the variable powers, such as ax^n + bx^(n-1) + ... + k.
Monomial consists of a single term, for example, 4x.
Binomial is made up of two distinct terms, like 3x^2 + 2.
Trinomial contains three terms, exemplified by x^2 + 5x + 6.
A Multinomial features more than three terms.
The Degree of a Polynomial represents the highest exponent in the polynomial; for instance, 4x^3 + 2x^2 + 1 has a degree of 3.
Addition and Subtraction of polynomials involve combining like terms for simplification.
Multiplication between polynomials utilizes the distributive property, often applying the FOIL method for binomials.
Division can be executed through long division or synthetic division techniques.
Factorization breaks down a polynomial into a product of its factors, such as transforming x^2 - 9 into (x - 3)(x + 3).

The Rational Root Theorem helps identify possible rational solutions of a polynomial equation.
Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots when counting multiplicities.
The Remainder Theorem asserts that the remainder of the division of a polynomial f(x) by (x - a) equals f(a).
Graphing Polynomials allows visualization of key features like intercepts, turning points, and end behavior, influenced by the polynomial's degree and leading coefficient.