Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• These symbols usually represent quantities without fixed values, known as variables.
• Algebra is more general than arithmetic, which only deals with specific numbers.

Basic Concepts

• Variables: Symbols (usually letters) representing unknown or changing quantities.
• Constants: Fixed values that do not change in an expression or equation.
• Expressions: Combinations of variables, constants, and mathematical operations (like addition, subtraction, multiplication, division, exponents, etc.). They do not contain an equals sign.
• Equations: Mathematical statements that show the equality between two expressions. They contain an equals sign (=).
• Coefficients: A number multiplied by a variable in an algebraic expression.
• Terms: Parts of an expression or equation separated by addition or subtraction.

Operations

• Addition (+): Combining terms. Only like terms (terms with the same variable raised to the same power) can be added directly.
• Subtraction (-): Finding the difference between terms; like addition, only like terms can be subtracted directly.
• Multiplication ( or ·):* Multiplying terms together. Unlike addition and subtraction, any terms can be multiplied. When multiplying variables, their exponents are added.
• Division (/ or ÷): Dividing terms. When dividing variables, their exponents are subtracted.
• Exponents: Indicate repeated multiplication of a base. For example, x^2 means x * x.

Order of Operations

• The order in which operations are performed is crucial to obtaining the correct result. A common mnemonic is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Parentheses (or Brackets): Operations inside parentheses are performed first.
• Exponents: Powers and roots are evaluated next.
• Multiplication and Division: These are performed from left to right.
• Addition and Subtraction: These are performed from left to right.

Solving Equations

• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• The goal is to isolate the variable on one side of the equation.
• Addition/Subtraction Property of Equality: Adding or subtracting the same value from both sides of an equation maintains the equality.
• Multiplication/Division Property of Equality: Multiplying or dividing both sides of an equation by the same non-zero value maintains the equality.
• Simplifying Expressions: Before solving an equation, simplify both sides by combining like terms and using the distributive property.

Linear Equations

• Linear equations are equations where the highest power of the variable is 1. The general form is ax + b = c, where a, b, and c are constants and x is the variable.
• To solve a linear equation, isolate the variable by performing inverse operations.

Quadratic Equations

• Quadratic equations are equations where the highest power of the variable is 2. The general form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Factoring: Expressing the quadratic expression as a product of two linear factors. Set each factor equal to zero and solve for x.
• Quadratic Formula: A general formula for finding the solutions of a quadratic equation: x = (-b ± √(b^2 - 4ac)) / (2a).
• If b^2 - 4ac < 0, the equation has no real solutions, or two complex solutions.
• If b^2 - 4ac = 0, the equation has exactly one real solution.
• If b^2 - 4ac > 0, the equation has two distinct real solutions.
• Completing the Square: A method for converting a quadratic equation into the form (x + p)^2 = q, which can then be solved by taking the square root of both sides.

Systems of Equations

• A system of equations is a set of two or more equations with the same variables.
• The solution to a system of equations is the set of 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 (Addition/Subtraction): Multiply one or both equations by a constant so that the coefficients of one variable are the same or opposite. Then, add or subtract the equations to eliminate that variable.
• Graphical Method: Graph each equation on the coordinate plane. The point(s) where the graphs intersect represent the solution(s) to the system.

Inequalities

• Inequalities are mathematical statements that compare two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities is similar to solving equations, but there is one important difference: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Degree of a Polynomial: The highest power of the variable in the polynomial.
• Standard Form: Writing a polynomial with the terms in descending order of degree.
• Adding and Subtracting Polynomials: Combine like terms.
• Multiplying Polynomials: Use the distributive property to multiply each term of one polynomial by each term of the other polynomial.
• Factoring Polynomials: Expressing a polynomial as a product of simpler polynomials.

Factoring

• Factoring is the process of breaking down an algebraic expression into its constituent factors.
• Greatest Common Factor (GCF): Find the largest factor that divides all terms in the expression 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
• a^2 - 2ab + b^2 = (a - b)^2
• Factoring by Grouping: Group terms in pairs and factor out the GCF from each pair.

Rational Expressions

• Rational expressions are expressions that can be written as a fraction where the numerator and denominator are polynomials.
• Simplifying Rational Expressions: Factor the numerator and denominator and cancel out any common factors.
• Multiplying Rational Expressions: Multiply the numerators and multiply the denominators. Simplify the resulting expression.
• Dividing Rational Expressions: Multiply by the reciprocal of the divisor.
• Adding and Subtracting Rational Expressions: Find a common denominator and combine the numerators.

Radicals

• A radical is a mathematical expression involving a root, such as a square root, cube root, etc.
• Simplifying Radicals: Find the largest perfect square (or cube, etc.) that is a factor of the radicand (the number under the radical sign) and take its root.
• Adding and Subtracting Radicals: Combine like radicals (radicals with the same radicand and index).
• Multiplying Radicals: Multiply the coefficients and multiply the radicands.
• Rationalizing the Denominator: Eliminate radicals from the denominator of a fraction by multiplying the numerator and denominator by a suitable expression (usually the conjugate of the denominator).

Functions

• A function is a relation between a set of inputs (called the domain) and a set of permissible outputs (called the range) with the property that each input is related to exactly one output.
• Function Notation: f(x) represents the output of the function f for the input x.
• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values that the function can produce.
• Linear Functions: Functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: Functions of the form f(x) = ax^2 + bx + c.
• Graphing Functions: Plotting points (x, f(x)) on the coordinate plane to visualize the function.