Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols
• These symbols represent quantities without fixed values, known as variables
• Algebra is fundamental to various fields including science, engineering, economics, and computer science

Basic Algebraic Concepts

• Variables are symbols (usually letters) representing unknown values or quantities that can change
• Constants are fixed values that do not change
• Coefficients are numerical or constant quantities placed before and multiplying the variable in an algebraic expression (e.g., in 3x, 3 is the coefficient)
• Expressions are combinations of variables, constants, and arithmetic operations (e.g., 3x + 5)
• Equations are statements showing the equality between two expressions (e.g., 3x + 5 = 14)

Algebraic Operations

• Addition is combining terms: like terms (terms with the same variable raised to the same power) can be added or subtracted (e.g., 3x + 2x = 5x)
• Subtraction is removing or reducing terms, similar to addition, only affecting like terms (e.g., 5y - 2y = 3y)
• Multiplication is distributing one term over another: terms can always be multiplied regardless of whether they are like terms (e.g., 2 * 3x = 6x, x * y = xy)
• Division is splitting terms: algebraic fractions involve dividing one algebraic expression by another (e.g., 6x / 2 = 3x)

Laws of Exponents

• Product of powers: when multiplying like bases, add the exponents (x^m * x^n = x^(m+n))
• Quotient of powers: when dividing like bases, subtract the exponents (x^m / x^n = x^(m-n))
• Power of a power: when raising a power to a power, multiply the exponents ((x^m)^n = x^(m*n))
• Power of a product: the power of a product is the product of the powers ((xy)^n = x^n * y^n)
• Power of a quotient: the power of a quotient is the quotient of the powers ((x/y)^n = x^n / y^n)
• Negative exponent: a negative exponent indicates a reciprocal (x^-n = 1/x^n)
• Zero exponent: anything raised to the power of zero is one (except 0) (x^0 = 1, x != 0)

Simplifying Algebraic Expressions

• Combining like terms: add or subtract terms that have the same variable raised to the same power (e.g., 5x + 3x - 2x = 6x)
• Distributive property: multiply a term by each term inside parentheses (a(b + c) = ab + ac)
• Factoring: express an expression as a product of its factors (e.g., x^2 + 5x + 6 = (x + 2)(x + 3))
• Expanding: remove parentheses by applying the distributive property (e.g., (x + 1)(x + 2) = x^2 + 3x + 2)

Solving Linear Equations

• Linear equations involve a single variable raised to the first power
• Isolate the variable by performing the same operations on both sides of the equation
• Use addition/subtraction to move constants or variables
• Use multiplication/division to remove coefficients

Solving Quadratic Equations

• Quadratic equations have the form ax^2 + bx + c = 0, where a != 0
• Factoring: factor the quadratic expression into two binomials and set each factor to zero (e.g., x^2 - 4 = (x - 2)(x + 2), so x = 2 or x = -2)
• Quadratic formula: use the formula x = (-b ± sqrt(b^2 - 4ac)) / (2a)
• Completing the square: rewrite the quadratic equation in the form (x - h)^2 = k to solve for x

Systems of Equations

• A system of equations is a set of two or more equations with the same variables
• Solving a system means finding values for the variables that satisfy all equations simultaneously
• Substitution: solve one equation for one variable and substitute that expression into the other equation
• Elimination: add or subtract multiples of the equations to eliminate one variable
• Graphing: plot the equations on a coordinate plane and find the points of intersection, which represent the solutions

Inequalities

• Inequalities are mathematical statements comparing two expressions using symbols like <, >, <=, or >=
• Solving inequalities involves finding the range of values that satisfy the inequality
• Perform the same operations on both sides, but reverse the inequality sign when multiplying or dividing by a negative number
• Interval notation is used to express the solution set (e.g., x > 3 is written as (3, ∞))

Polynomials

• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents
• Terms are parts of a polynomial separated by addition or subtraction
• Degree of a polynomial is the highest power of the variable in the polynomial
• Standard form arranges terms in descending order of degree
• Addition/Subtraction: combine like terms
• Multiplication: use the distributive property
• Division: polynomial 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
• Sum/Difference of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) or a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Grouping: group terms and factor out common factors to simplify

Rational Expressions

• Rational expressions are fractions where the numerator and denominator are polynomials
• Simplifying Rational Expressions: factor the numerator and denominator and cancel common factors
• Multiplication: multiply the numerators and the denominators
• Division: multiply by the reciprocal of the divisor
• Addition/Subtraction: find a common denominator and combine the numerators

Radicals

• Radicals are expressions involving roots, such as square roots, cube roots, etc.
• Simplifying Radicals: factor out perfect squares/cubes from the radicand (the expression under the radical sign)
• Operations with Radicals: combine like radicals (radicals with the same index and radicand)
• Rationalizing the Denominator: remove radicals from the denominator by multiplying the numerator and denominator by a suitable expression

Functions

• A function is 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
• Notation: f(x) represents the output of the function f for the input x
• Domain is the set of all possible input values (x)
• Range is the set of all possible output values (f(x))
• Linear Functions: have the form f(x) = mx + b, where m is the slope and b is the y-intercept
• Quadratic Functions: have the form f(x) = ax^2 + bx + c, and their graphs are parabolas

Graphing

• Linear Equations: plot two points and draw a line through them (y = mx + b)
• Inequalities: shade the region of the coordinate plane that satisfies the inequality
• Systems of Equations: find the intersection point(s) of the graphs of the equations
• Functions: plot points and connect them to visualize the function's behavior