Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• It provides a framework for expressing relationships and solving equations
• Variables are symbols (usually letters) that represent unknown or changing values
• Constants are fixed values that do not change
• Expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, division)
• Equations are statements that two expressions are equal
• Formulas are equations that express a relationship between two or more variables

Basic Operations

• Addition: Combining two or more terms
• Subtraction: Finding the difference between two terms
• Multiplication: Repeated addition of a term
• Division: Splitting a term into equal parts
• Exponents: Represents repeated multiplication of a base by itself
• Radicals: Represents the root of a number (e.g., square root, cube root)

Order of Operations

• Parentheses (or brackets)
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)
• PEMDAS is a common mnemonic for remembering the order

Combining Like Terms

• Like terms have the same variable(s) raised to the same power
• Combine like terms by adding or subtracting their coefficients
• For example, 3x + 5x = 8x

Distributive Property

• Distribute a term by multiplying it by each term inside parentheses
• a(b + c) = ab + ac

Factoring

• Factoring is the reverse of distribution
• Involves breaking down an expression into factors that, when multiplied together, give the original expression
• Example: xy + xz = x(y + z)

Solving Linear Equations

• Goal: Isolate the variable on one side of the equation
• Use inverse operations to undo operations performed on the variable
• Maintain balance by performing the same operation on both sides of the equation
• Example: 2x + 3 = 7
• Subtract 3 from both sides: 2x = 4
• Divide both sides by 2: x = 2

Solving Linear Inequalities

• Similar to solving equations, but with an inequality sign (<, >, ≤, ≥)
• When multiplying or dividing by a negative number, reverse the inequality sign
• Example: -2x < 6
• Divide both sides by -2 (and reverse the inequality): x > -3

Systems of Linear Equations

• A set of two or more linear equations with the same variables
• Goal: Find the values of the variables that satisfy all equations simultaneously
• Methods for solving:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation
  • Elimination (or Addition): Add or subtract the equations to eliminate one variable
  • Graphing: Graph each equation and find the point of intersection

Exponents

• An exponent indicates how many times a base is multiplied by itself
• x^n means x multiplied by itself n times
• Laws of exponents:
  • Product of powers: x^m * x^n = x^(m+n)
  • Quotient of powers: x^m / x^n = x^(m-n)
  • Power of a power: (x^m)^n = x^(m*n)
  • Power of a product: (xy)^n = x^n * y^n
  • Power of a quotient: (x/y)^n = x^n / y^n
  • Negative exponent: x^(-n) = 1 / x^n
  • Zero exponent: x^0 = 1 (if x ≠ 0)

Polynomials

• An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents
• Examples: 3x^2 + 2x - 1, 5y^3 - 7
• Not polynomials: 3x^(-2) + 2, 5/x + 1
• Degree of a polynomial: The highest power of the variable in the polynomial
• Types of polynomials:
  • Monomial: One term (e.g., 5x^2)
  • Binomial: Two terms (e.g., 2x + 3)
  • Trinomial: Three terms (e.g., x^2 - 4x + 7)

Operations with Polynomials

• Adding and Subtracting: Combine like terms
• Multiplying: Distribute each term of one polynomial to each term of the other polynomial (FOIL method for binomials)
• Dividing: Long division or synthetic division

Factoring Polynomials

• Greatest Common Factor (GCF): Find the largest factor common to all terms and factor it out
• Difference of Squares: a^2 - b^2 = (a + b)(a - b)
• Perfect Square Trinomials: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2
• Quadratic Trinomials: Factor into two binomials (trial and error or using the quadratic formula if unsolvable)

Quadratic Equations

• An equation of the form ax^2 + bx + c = 0, where a ≠ 0
• Methods for solving:
  • Factoring: Factor the quadratic expression and set each factor equal to zero
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
  • Completing the Square: Manipulate the equation to form a perfect square trinomial
• Discriminant (b^2 - 4ac): Determines the nature of the roots
  • If b^2 - 4ac > 0: Two distinct real roots
  • If b^2 - 4ac = 0: One real root (a repeated root)
  • If b^2 - 4ac < 0: Two complex roots

Radicals

• A radical expression contains a root (e.g., square root, cube root)
• Simplifying radicals: Factor out perfect squares (or cubes, etc.) from under the radical
• Operations with radicals:
  • Adding and Subtracting: Combine like radicals (radicals with the same index and radicand)
  • Multiplying: Multiply the radicands and simplify
  • Dividing: Rationalize the denominator (multiply the numerator and denominator by a radical that eliminates the radical from the denominator)

Rational Expressions

• A fraction in which the numerator and/or the denominator are polynomials
• Simplifying rational expressions: Factor the numerator and denominator and cancel common factors
• Operations with rational expressions:
  • Adding and Subtracting: Find a common denominator and combine the numerators
  • Multiplying: Multiply the numerators and multiply the denominators
  • Dividing: Multiply by the reciprocal of the divisor

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
• Types of functions:
  • Linear functions: f(x) = mx + b (m is the slope, b is the y-intercept)
  • Quadratic functions: f(x) = ax^2 + bx + c
  • Polynomial functions: f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
  • Rational functions: f(x) = p(x) / q(x), where p(x) and q(x) are polynomials
• Domain: The set of all possible input values (x-values)
• Range: The set of all possible output values (f(x)-values)

Graphing

• Visual representation of algebraic relationships
• Linear equations graph as straight lines
• Quadratic equations graph as parabolas
• Slope: Measure of the steepness of a line (rise over run)
• Intercepts: Points where the graph crosses the x-axis (x-intercept) or y-axis (y-intercept)