Introduction to Algebra

• Algebra is a branch of mathematics that uses symbols and rules to manipulate those symbols.
• It acts as a unifying thread throughout almost all areas of mathematics.
• Algebra encompasses various topics, ranging from solving equations to studying abstract concepts like groups, rings, and fields.

Basic Algebraic Operations

• Addition, subtraction, multiplication, division, exponentiation, and root extraction are considered basic algebraic operations.
• Algebraic operations should follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Variables and Expressions

• A variable is a symbol, usually a letter, that stands for a value or quantity that can change.
• An algebraic expression combines variables, numbers, and algebraic operations.

Equations and Inequalities

• An equation states that two expressions are equal.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• An inequality compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
• Solving an inequality involves determining the range of values for the variable(s) that satisfy the inequality.

Polynomials

• A polynomial includes variables and coefficients and uses the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• A polynomial in one variable can be written as a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are coefficients, and n is a non-negative integer representing the polynomial's degree.

Operations with Polynomials

• Polynomials can undergo addition, subtraction, multiplication, and division.
• Adding or subtracting polynomials means combining like terms, which have the same variable and exponent.
• Multiplying polynomials involves using the distributive property to multiply each term of one polynomial by each term of the other polynomial.
• Polynomial division can be carried out using long division or synthetic division.

Factoring Polynomials

• Factoring a polynomial involves rewriting it as a product of simpler polynomials or factors.
• Common factoring methods include factoring out the greatest common factor (GCF), factoring by grouping, and using special product formulas like the difference of squares and perfect square trinomials.

Linear Equations

• A linear equation is an equation where the highest power of the variable is 1.
• Linear equations can be expressed in the form ax + b = 0, where a and b are constants, and x is the variable.

Solving Linear Equations

• Solving linear equations requires isolating the variable on one side of the equation.
• This isolation is achieved through inverse operations such as addition/subtraction and multiplication/division to reverse the operations performed on the variable.

Systems of Linear Equations

• A system of linear equations includes two or more linear equations that share the same variables.
• Solving such a system means finding the values of the variables that satisfy all equations simultaneously.
• Common solution methods are substitution, elimination, and graphing.

Quadratic Equations

• A quadratic equation is defined as an equation in which the highest power of the variable is 2.
• Quadratic equations are expressed as ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Solving Quadratic Equations

• Quadratic equations can be solved through factoring, completing the square, or using the quadratic formula.
• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).
• The discriminant (b^2 - 4ac) helps determine the nature of the roots:
  • If b^2 - 4ac > 0, the equation has two distinct real roots.
  • If b^2 - 4ac = 0, the equation has one real root (a repeated root).
  • If b^2 - 4ac < 0, the equation has two complex roots.

Exponents and Radicals

• An exponent indicates how many times a base number is multiplied by itself.
• In the expression x^n, x is the base, and n is the exponent.

Rules 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 (provided x ≠ 0)

Radicals

• A radical (√) symbolizes the root of a number.
• The nth root of x is a number y such that y^n = x.
• √x represents the square root of x.
• ∛x represents the cube root of x.

Simplifying Radicals

• Radicals are simplified by factoring out perfect squares, cubes, etc., from the radicand.
• For instance, √12 = √(4 * 3) = √4 * √3 = 2√3.

Rationalizing the Denominator

• This process removes radicals from the denominator of a fraction.
• It involves multiplying both the numerator and the denominator by an expression that eliminates the radical in the denominator.

Functions

• A function is a relation where each input corresponds to exactly one output.
• The input is called the argument or independent variable.
• The output is called the value or dependent variable.

Types of Functions

• Linear functions (y = mx + b)
• Quadratic functions (y = ax^2 + bx + c)
• Polynomial functions
• Rational functions (ratio of two polynomials)
• Exponential functions (y = a^x)
• Logarithmic functions (y = log_a(x))
• Trigonometric functions (sin(x), cos(x), tan(x), etc.)

Function Notation

• Functions are denoted by symbols like f, g, or h.
• The value of function f at an input x is written as f(x).

Graphing Functions

• A function's graph includes all points (x, f(x)) on the coordinate plane.
• The graph helps visualize the function's behavior.

Coordinate Geometry

• Coordinate geometry studies geometric figures using a coordinate system.

Coordinate Plane

• It includes two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• Points are represented as ordered pairs (x, y), where x is the x-coordinate (abscissa), and y is the y-coordinate (ordinate).

Distance Formula

• The distance between points (x_1, y_1) and (x_2, y_2) is √((x_2 - x_1)^2 + (y_2 - y_1)^2).

Slope

• The slope of a line through points (x_1, y_1) and (x_2, y_2) is (y_2 - y_1) / (x_2 - x_1).
• The slope indicates the rate of change of the y-coordinate relative to the x-coordinate.

Equations of Lines

• Slope-intercept form: y = mx + b, where m is the slope, and b is the y-intercept.
• Point-slope form: y - y_1 = m(x - x_1), where m is the slope, and (x_1, y_1) is a point on the line.
• Standard form: Ax + By = C, where A, B, and C are constants.