Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols and rules to manipulate them.
• Symbols in algebra represent quantities without fixed values, known as variables.
• Algebra serves as a generalization of arithmetic.
• Mathematical relationships can be formulated and equations solved using algebra.

Basic Operations

• Addition, subtraction, multiplication, and division are the four basic operations.
• These operations form the basis for algebraic manipulation.
• Understanding the properties of these operations is key to simplifying expressions and solving equations.

Variables and Constants

• Variables are symbols, usually letters, that represent unknown or changing values, examples include x, y, and z.
• Constants are fixed values that do not change; examples include 2, 7, and π.
• Coefficients are numbers multiplied by variables (e.g., in 3x, 3 is the coefficient).

Algebraic Expressions

• Algebraic expressions are made up of variables, constants, and operations; an example is 3x + 2y - 5.
• Terms are the individual components of an expression, separated by + or - signs; in 3x + 2y - 5, the terms are 3x, 2y, and -5.
• Like terms have identical variables raised to the same powers; for example, 3x and 5x.
• Unlike terms have different variables or different powers; for example, 3x and 2y.

Simplifying Expressions

• Combining like terms means adding or subtracting the coefficients of like terms; for example, 3x + 5x = 8x.
• Distributive property: a(b + c) = ab + ac.
• Factoring involves expressing an expression as a product of its factors; for example, x^2 + 2x = x(x + 2).

Equations

• An equation is a statement showing the equality of two expressions; for example, 2x + 3 = 7.
• The objective is to determine the value(s) of the variable(s) that make the equation true.
• Solutions or roots are the values that satisfy the equation.

Solving Linear Equations

• Linear equations contain variables raised to the first power.
• The goal is to isolate the variable on one side of the equation.
• Inverse operations are used to reverse the operations performed on the variable.
• Balance must be maintained by performing the same operation on both sides of the equation.

Examples of Solving Linear Equations

• To solve x + 5 = 9: Subtract 5 from both sides, resulting in x = 4.
• To solve 3x = 12: Divide both sides by 3, resulting in x = 4.
• To solve 2x + 3 = 9: Subtract 3 from both sides (2x = 6), and then divide by 2, resulting in x = 3.

Solving Systems of Linear Equations

• A system of linear equations involves two or more equations that share the same variables.
• Solutions must satisfy all equations in the system simultaneously.
• Substitution, elimination, and graphing can be used to solve these systems.

Substitution Method

• First, solve one of the equations for one variable.
• Then, substitute that expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Finally, substitute the value back into one of the original equations to find the value of the other variable.

Elimination Method

• Multiply one or both equations by a constant so that one variable's coefficients are opposites.
• Add the equations together to eliminate the variable with opposite coefficients.
• Solve for the remaining variable.
• Substitute the value back into one of the original equations to find the value of the eliminated variable.

Quadratic Equations

• A quadratic equation is in the form ax^2 + bx + c = 0, where a ≠ 0.
• Factoring, completing the square, and the quadratic formula can be used to find solutions.

Factoring Quadratic Equations

• Express the quadratic expression as a product of two binomials.
• Set each factor equal to zero and solve for x.
• Example: x^2 - 5x + 6 = (x - 2)(x - 3) = 0, therefore x = 2 or x = 3.

Quadratic Formula

• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
• The discriminant (b^2 - 4ac) determines the nature of the roots:
  • There are two distinct real roots if b^2 - 4ac > 0.
  • There is one real root (a repeated root) when b^2 - 4ac = 0.
  • There are two complex roots if b^2 - 4ac < 0.

Functions

• Functions relate a set of inputs to a set of permissible outputs, where each input corresponds to exactly one output.
• The input is called the argument, and the output is the function's value.
• Functions are represented by symbols such as f, g, or h.

Function Notation

• f(x) is the value of the function f at x.
• x is the independent variable, while f(x) is the dependent variable.
• Example: If f(x) = x^2 + 1, then f(3) = 3^2 + 1 = 10.

Domain and Range

• The domain is the set of all possible input values (x-values) for a function.
• The range is the set of all possible output values (f(x)-values) for a function.

Composition of Functions

• The composition of functions f and g, written as (f ∘ g)(x) or f(g(x)), is the function that applies g to x and then applies f to the result.
• Example: If f(x) = x + 1 and g(x) = x^2, then f(g(x)) = f(x^2) = x^2 + 1.

Inequalities

• Inequalities compare two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities means finding the range of values that satisfy the inequality.

Solving Linear Inequalities

• Solving linear inequalities is similar to solving linear equations, but multiplying or dividing by a negative number reverses the inequality sign.
• Example: To solve -2x < 6, divide by -2 and reverse the sign: x > -3.

Absolute Value

• The absolute value of a number is its distance from zero, written as |x|.
• |x| is always non-negative.
• |x| = x if x ≥ 0, and |x| = -x if x < 0.

Solving Absolute Value Equations

• To solve |x| = a, consider two cases: x = a and x = -a.
• Example: To solve |x - 2| = 3, solve x - 2 = 3 or x - 2 = -3, so x = 5 or x = -1.

Exponents

• Exponents indicate how many times to multiply a base by itself.
• x^n means multiplying x by itself n times.
• Laws of exponents provide rules for manipulating expressions with exponents.

Laws 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).
• Power of a product: (xy)^a = x^a * y^a.
• Power of a quotient: (x/y)^a = x^a / y^a.
• Zero exponent: x^0 = 1 (if x ≠ 0).
• Negative exponent: x^(-a) = 1 / x^a.

Radicals

• A radical (√) represents the root of a number.
• √x is the square root of x.
• ∛x is the cube root of x.

Simplifying Radicals

• Radicals are simplified by factoring out perfect squares (or cubes, etc.).
• Example: √12 = √(4 * 3) = √4 * √3 = 2√3.

Rationalizing the Denominator

• Removing radicals from the denominator of a fraction is called rationalizing the denominator.
• Multiply the numerator and denominator by a suitable expression.
• Example: To rationalize 1/√2, multiply by √2/√2 to get √2/2.

Polynomials

• Polynomials are expressions with variables and coefficients, involving addition, subtraction, multiplication, and non-negative integer exponents of variables.
• Examples of polynomials: x^2 + 3x - 4 and 5y^3 - 2y + 1.

Operations with Polynomials

• To add or subtract polynomials, combine like terms.
• To multiply polynomials, use the distributive property.
• To divide polynomials, use polynomial long division or synthetic division.

Factoring Polynomials

• Factoring is expressing a polynomial as a product of simpler polynomials or monomials.
• Common techniques include: finding the Greatest Common Factor (GCF), using the difference of squares, recognizing perfect square trinomials, and grouping.

Graphing Linear Equations

• Linear equations are graphically represented as straight lines on a coordinate plane.
• The standard form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• Slope measures the steepness of the line.
• The y-intercept indicates where the line intersects the y-axis.