Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• It is a generalization of arithmetic, where specific numerical values are used

Basic Concepts

• Variable: A symbol (usually a letter) that represents an unknown or unspecified value
• Constant: A fixed value that does not change
• Expression: A combination of variables, constants, and mathematical operations (+, -, ×, ÷)

Operators

• Addition (+): Combines two or more terms
• Subtraction (-): Finds the difference between two terms
• Multiplication (× or *): Repeats a term a certain number of times
• Division (÷ or /): Splits a term into equal parts
• Exponentiation (^): Raises a term to a power

Equations

• An equation is a statement that two expressions are equal
• It contains an equals sign (=)
• The goal is typically to find the value(s) of the variable(s) that make the equation true
• Linear Equation: An equation where the highest power of the variable is 1

Solving Linear Equations

• Isolate the variable on one side of the equation
• Use inverse operations to undo the operations performed on the variable
• Addition and subtraction are inverse operations
• Multiplication and division are inverse operations
• Example: Solve for x: 2x + 3 = 7
  • Subtract 3 from both sides: 2x = 4
  • Divide both sides by 2: x = 2
• The solution is x = 2

Inequalities

• Inequalities compare two expressions using inequality symbols
  • Less than (<)
  • Greater than (>)
  • Less than or equal to (≤)
  • Greater than or equal to (≥)
• Solving inequalities is similar to solving equations, but there are a few key differences
• Multiplying or dividing both sides by a negative number reverses the inequality sign
• Example: Solve for x: -3x < 9
  • Divide both sides by -3 (and reverse the inequality sign): x > -3
• The solution is x > -3

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Terms: Parts of a polynomial separated by addition or subtraction
• Coefficients: Numerical factors of the terms
• Degree: The highest power of the variable in a polynomial
• Monomial: A polynomial with one term
• Binomial: A polynomial with two terms
• Trinomial: A polynomial with three terms

Operations with Polynomials

• Adding and Subtracting: Combine like terms (terms with the same variable and exponent)
• Multiplying: Use the distributive property to multiply each term in one polynomial by each term in the other polynomial
• Example: (x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6

Factoring Polynomials

• Factoring is the process of writing a polynomial as a product of simpler polynomials
• Common Factoring Techniques:
  • Greatest Common Factor (GCF): Find the largest factor that divides all terms
  • Difference of Squares: a² - b² = (a + b)(a - b)
  • Perfect Square Trinomial: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
  • Factoring by Grouping: Group terms and factor out common factors
• Example: Factor x² - 4
  • Recognize as a difference of squares: x² - 2²
  • Apply the formula: (x + 2)(x - 2)

Quadratic Equations

• A quadratic equation is a polynomial equation of the second degree
• Standard Form: ax² + bx + c = 0, where a ≠ 0
• Solving Quadratic Equations:
  • Factoring: Factor the quadratic expression and set each factor equal to zero
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
  • Completing the Square: Rewrite the equation in the form (x + p)² = q and solve for x
• Discriminant: The part of the quadratic formula under the square root (b² - 4ac)
  • If b² - 4ac > 0: Two distinct real solutions
  • If b² - 4ac = 0: One real solution (a repeated root)
  • If b² - 4ac < 0: Two complex solutions
• Vertex Form: a(x-h)^2 +k = 0, where the vertex is located at the point (h, k)

Systems of Equations

• A system of equations is a set of two or more equations with the same variables
• Solving Systems of Equations:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation
  • Elimination: Add or subtract the equations to eliminate one variable
• Types of Solutions:
  • Unique Solution: The lines intersect at one point
  • No Solution: The lines are parallel and do not intersect
  • Infinite Solutions: The lines are the same (coincident)

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range) with the property that each input is related to exactly one output
• Notation: f(x), where x is the input and f(x) is the output
• Linear Function: A function whose graph is a straight line (f(x) = mx + b)
• Quadratic Function: A function whose graph is a parabola (f(x) = ax² + bx + c)

Exponents and Radicals

• Exponent Rules:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(m*n)
  • (ab)^n = a^n * b^n
  • a^0 = 1 (if a ≠ 0)
  • a^(-n) = 1 / a^n
• Radicals:
  • √a is the square root of a
  • n√a is the nth root of a
  • a^(1/n) = n√a
• Rationalizing the Denominator: Eliminate radicals from the denominator of a fraction by multiplying by a suitable form of 1

Logarithms

• A logarithm is the inverse operation to exponentiation
• Notation: log_b(x) = y if and only if b^y = x
• b is the base of the logarithm
• Common Logarithm: Base 10 (log₁₀(x) or simply log(x))
• Natural Logarithm: Base e (logₑ(x) or ln(x)), where e is approximately 2.71828
• Logarithm Rules:
  • log_b(mn) = log_b(m) + log_b(n)
  • log_b(m/n) = log_b(m) - log_b(n)
  • log_b(m^p) = p * log_b(m)
  • log_b(1) = 0
  • log_b(b) = 1
  • Change of Base Formula: log_a(x) = log_b(x) / log_b(a)

Complex Numbers

• A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)
• a is the real part and b is the imaginary part
• Operations with Complex Numbers:
  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
  • Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
  • Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
  • Division: Multiply the numerator and denominator by the conjugate of the denominator

Absolute value

• Represents the distance a number is from zero on the number line
• Denoted as |x|, where x is a real number
• |x| = x if x ≥ 0, and |x| = -x if x < 0
• Always non-negative
• Geometrically, |x - a| represents the distance between points x and a on the number line.