Algebraic Equations: Definition and Types

Study Notes

An algebraic equation is an equation involving variables and constants, where the variables are raised to integral powers.
Types of algebraic equations:
- Linear equations: degree of the variable is 1 (e.g., 2x + 3 = 0)
- Quadratic equations: degree of the variable is 2 (e.g., x^2 + 4x + 4 = 0)
- Cubic equations: degree of the variable is 3 (e.g., x^3 - 2x^2 - 7x + 1 = 0)
- Polynomial equations: degree of the variable is greater than 3 (e.g., x^4 - 3x^3 - 2x^2 + 5x - 1 = 0)

Methods for solving algebraic equations:
- Factoring: expressing the equation as a product of simpler equations (e.g., x^2 + 5x + 6 = (x + 3)(x + 2) = 0)
- Quadratic Formula: for quadratic equations, x = (-b ± √(b^2 - 4ac)) / 2a
- Synthetic Division: a method for finding the roots of a polynomial equation
- Graphical Method: using graphs to find the roots of an equation

Divisibility rules:
- A number is divisible by 2 if its last digit is even
- A number is divisible by 3 if the sum of its digits is divisible by 3
- A number is divisible by 5 if its last digit is 0 or 5
Prime numbers: positive integers greater than 1 that are divisible only by 1 and themselves
Properties of prime numbers:
- Prime numbers are always odd, except for 2
- The sum of any two prime numbers is always even
- Prime numbers are building blocks of all other numbers (Fundamental Theorem of Arithmetic)

Greatest Common Divisor (GCD): the largest positive integer that divides both numbers without leaving a remainder
Least Common Multiple (LCM): the smallest positive integer that is a multiple of both numbers
Methods for finding GCDs and LCMs:
- Euclidean Algorithm: a method for finding the GCD of two numbers
- Prime Factorization: expressing a number as a product of its prime factors to find the GCD and LCM

Algebraic equations involve variables and constants, with variables raised to integral powers.
Types of algebraic equations include:
- Linear equations, where the degree of the variable is 1, such as 2x + 3 = 0.
- Quadratic equations, where the degree of the variable is 2, such as x^2 + 4x + 4 = 0.
- Cubic equations, where the degree of the variable is 3, such as x^3 - 2x^2 - 7x + 1 = 0.
- Polynomial equations, where the degree of the variable is greater than 3, such as x^4 - 3x^3 - 2x^2 + 5x - 1 = 0.

Methods for solving algebraic equations include:
- Factoring, which involves expressing the equation as a product of simpler equations.
- The Quadratic Formula, x = (-b ± √(b^2 - 4ac)) / 2a, for solving quadratic equations.
- Synthetic Division, a method for finding the roots of a polynomial equation.
- The Graphical Method, which uses graphs to find the roots of an equation.

Divisibility rules:
- A number is divisible by 2 if its last digit is even.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 5 if its last digit is 0 or 5.
Prime numbers are positive integers greater than 1 that are divisible only by 1 and themselves.
Properties of prime numbers include:
- Prime numbers are always odd, except for 2.
- The sum of any two prime numbers is always even.
- Prime numbers are the building blocks of all other numbers (Fundamental Theorem of Arithmetic).

The Greatest Common Divisor (GCD) is the largest positive integer that divides both numbers without leaving a remainder.
The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of both numbers.
Methods for finding GCDs and LCMs include:
- The Euclidean Algorithm, a method for finding the GCD of two numbers.
- Prime Factorization, which involves expressing a number as a product of its prime factors to find the GCD and LCM.