Introduction to Algebra

• Algebra manipulates symbols by using rules
• Symbols are variables representing quantities without fixed values
• Algebra is more general than arithmetic

Basic Algebraic Operations

• Addition, subtraction, multiplication, and division are fundamental
• Operations are applicable to both numbers and variables
• Algebraic expressions are a combination of numbers, variables, and operations

Variables and Constants

• Variables are symbols that represent unknown or changing values
• Constants have fixed numerical values in an expression

Algebraic Expressions

• Made up of variables, constants, and algebraic operations
• Examples include: 3x + 2, y^2 - 5, (a + b) / c
• Terms are separated by + or - signs

Coefficients

• Numerical factor of a term that contains a variable
• 3 is the coefficient of x in 3x

Equations

• Equations are mathematical statements asserting equality between two expressions
• Contains an equals sign (=)
• For example: 2x + 5 = 11

Solving Equations

• Process determines the value(s) of variables to satisfy the equation
• Isolation of the variable is achieved by performing identical operations on both sides
• Balance must be maintained to preserve equality

Linear Equations

• The highest variable power is 1
• The general form is ax + b = 0, where a and b are constants, and a ≠ 0
• Isolation of the variable is achieved using inverse operations

Quadratic Equations

• The highest variable power is 2
• The general form is ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0
• Factoring, completing the square, or using the quadratic formula can solve these equations

Factoring

• A quadratic expression is expressed as the product of two linear expressions
• For example: x^2 - 4 = (x + 2)(x - 2)
• Useful in finding roots (solutions) of the quadratic equation

Quadratic Formula

• Solutions are found when factoring is not straightforward
• Given by: x = (-b ± √(b^2 - 4ac)) / (2a)
• Where a, b, and c are coefficients from the quadratic equation ax^2 + bx + c = 0

Completing the Square

• Technique converts a quadratic equation into a perfect square form
• Allows for simple extraction of the solutions
• A constant is added and subtracted to complete the square

Systems of Equations

• A set of two or more equations using the same variables
• The solution is a set of values for the variables satisfying all equations
• Solving methods consist of substitution, elimination, and graphing

Substitution Method

• One equation is solved for one variable, then substitution of that expression into the other equation takes place
• The result is a single equation with one variable, which can then be solved
• To find the other variable, the value is substituted back

Elimination Method

• Equations are either added or subtracted to eliminate one of the variables
• Multiplication of one or both equations by constants makes the variable coefficients equal or opposite
• Solving is for the remaining variable, with substitution to find the other

Graphing Method

• Each equation is graphed on the coordinate plane
• System of equations' solutions are represented by intersection points
• Solutions may be visualized this way

Inequalities

• Shows the relationship between unequal expressions
• Symbols used: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to)
• Solving means finding the value ranges that satisfy the inequality

Linear Inequalities

• Similar to linear equations, but with an inequality sign
• For example: 2x + 3 < 7
• The solution is a value range for x

Solving Inequalities

• Like solving equations, except multiplying or dividing by a negative number reverses the inequality sign
• The solution set is found by isolating the variable

Graphing Inequalities

• The solution set is represented on a number line
• Use open circles for < and > and closed circles for ≤ and ≥
• Shade the region that satisfies the inequality

Polynomials

• Algebraic expression with variables and coefficients using addition, subtraction, multiplication, and non-negative exponents
• For example: 4x^3 - 2x^2 + x - 5
• Terms are separated by either addition or subtraction

Degree of a Polynomial

• The highest variable power in the polynomial
• For example: 4x^3 - 2x^2 + x - 5 is 3

Operations with Polynomials

• Polynomials can be added, subtracted, multiplied, and divided
• When adding or subtracting, like terms are combined

Factoring Polynomials

• A polynomial is expressed as a product of simpler polynomials
• Factoring out the greatest common factor, difference of squares, and perfect square trinomials are techniques used

Rational Expressions

• A fraction where the numerator and denominator are polynomials
• For example: (x + 1) / (x - 2)
• Involves simplifying, adding, subtracting, multiplying, and dividing in its operations

Simplifying Rational Expressions

• Divide both the numerator and denominator by their greatest common factor
• Is restricted because the Denominator cannot be zero

Operations with Rational Expressions

• Multiplication: multiply numerators and denominators
• Division: multiply by the reciprocal of the divisor
• Addition and Subtraction: Find a common denominator and combine numerators

Exponents and Radicals

• Exponents represent repeated multiplication
• Example: x^3 = x * x * x

Rules of Exponents

• Product of powers: x^a * x^b = x^(a+b)
• Quotient of powers: x^a / x^b = x^(a-b)
• Power of a power: (x^a)^b = x^(ab)
• Negative exponent: x^(-a) = 1 / x^a
• Zero exponent: x^0 = 1 (x ≠ 0)

Radicals

• Represents the root of a number
• √x represents the square root of x
• Radicals can be expressed as fractional exponents

Simplifying Radicals

• Extract perfect square factors from the radicand
• To remove radicals from the denominator, rationalize it

Logarithms

• Inverse operation to exponentiation
• If b^y = x, then log_b(x) = y
• b is the base of the logarithm

Properties of Logarithms

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x/y) = log_b(x) - log_b(y)
• Power rule: log_b(x^p) = p * log_b(x)
• Change of base formula: log_a(x) = log_b(x) / log_b(a)

Functions

• A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• Represented as f(x), where x is the input and f(x) is the output

Domain and Range

• Domain: The set of all possible input values (x)
• Range: The set of all possible output values (f(x))
• Algebra is useful and appears in many fields of studies and applications