Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and their manipulation.
• Symbols in algebra represent quantities without fixed values, known as variables.
• Algebra serves as a unifying element in nearly all areas of mathematics.

Variables

• A variable, typically a letter, symbolizes a value that is either unknown or subject to change.
• Variables articulate relationships between quantities.
• Common variable letters include x, y, z, a, b, c, and n.

Constants

• A constant is a value that remains fixed and does not change.
• Numbers, such as 2, -5, and π, are constants.

Expressions

• An algebraic expression is created by combining variables, constants, and algebraic operations like addition, subtraction, multiplication, division, and exponentiation.
• An example of an expression is 3x + 2y - 5.

Equations

• An equation is a statement asserting the equality of two expressions.
• Equations are characterized by containing an equals sign (=).
• An example of an equation is 3x + 2 = 11.

Terms

• A term constitutes a segment of an expression or equation, demarcated by a + or − sign.
• In the expression 4x − 7, the terms are 4x and −7.

Coefficients

• A coefficient is a numerical value that is multiplied by a variable.
• In the term 4x, the coefficient is 4.

Operators

• Operators are symbols denoting mathematical operations.
• Examples include +, −, ×, ÷, and ^ (exponentiation).

Basic Operations

• Addition (+): The process of combining two or more numbers or variables.
• Subtraction (−): The process of determining the difference between two numbers or variables.
• Multiplication (× or implied): The process of finding the product of two or more numbers or variables.
• Division (÷ or /): The process of splitting a number or variable into equal portions.
• Exponentiation (^): The process of raising a number or variable to a specified power.

Order of Operations

• The order of operations constitutes a set of rules stipulating the sequence in which mathematical operations must be executed.
• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is an acronym often employed to recall the correct order.

Simplifying Expressions

• Combining Like Terms: Combining terms that share the same variable raised to the same power through addition or subtraction of their coefficients.
• For example: 3x + 2x = 5x
• Distributive Property: Multiplying a number by each term enclosed within parentheses.
• For example: a(b + c) = ab + ac

Solving Equations

• Solving an equation aims to determine the value(s) of the variable(s) that render the equation true.
• Isolate the Variable: Employ inverse operations to isolate the variable on one side of the equation.
• Addition/Subtraction Principle: Maintaining equality by adding or subtracting the same number from both sides of an equation.
• Multiplication/Division Principle: Maintaining equality by multiplying or dividing both sides of an equation by the same non-zero number.

Linear Equations

• A linear equation is identified as an equation where the highest power of the variable is 1.
• Its standard form is expressed as ax + b = c, where a, b, and c represent constants.

Solving Linear Equations

• Inverse operations are used to isolate the variable.
• For example: 2x + 3 = 7
  • Subtract 3 from both sides to get 2x = 4.
  • Divide both sides by 2 to find x = 2.

Systems of Linear Equations

• A system of linear equations comprises two or more linear equations sharing the same variables.
• The solution to such a system is the set of values for the variables that satisfy all equations within the system.

Methods for Solving Systems of Linear Equations

• Substitution Method: Solving one equation for a single variable and substituting that expression into another equation.
• Elimination Method: Adding or subtracting multiples of equations to eliminate one of the variables.
• Graphing Method: Graphing each equation and identifying the point(s) of intersection.

Quadratic Equations

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

Methods for Solving Quadratic Equations

• Factoring: Expressing the quadratic expression as a product of two linear factors.
• Quadratic Formula: Applying the formula x = (-b ± √(b^2 - 4ac)) / (2a) to determine the solutions.
• Completing the Square: Rewriting the equation in the form (x - h)^2 = k and subsequently solving for x.

Factoring

• Factoring is the process of simplifying an algebraic expression into a product of simpler expressions.
• Common Techniques:
  • Greatest Common Factor (GCF): Identifying and factoring out the largest factor shared by all terms.
  • Difference of Squares: Applying the formula a^2 - b^2 = (a + b)(a - b).
  • Perfect Square Trinomials: Using the formulas a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2.

Polynomials

• Defined as expressions containing variables and coefficients, combined through addition, subtraction, multiplication, and non-negative integer exponents.
• General form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_i represents coefficients and n is a non-negative integer.

Operations with Polynomials

• Addition and Subtraction: Combination of like terms.
• Multiplication: The distributive property applies to multiply each term in one polynomial by each term in the other polynomial.

Exponents

• An exponent specifies the number of times a base is multiplied by itself.
• For example: x^n signifies x multiplied by itself n times.

Rules of Exponents

• Product of Powers: a^m * a^n = a^{m+n}
• Quotient of Powers: a^m / a^n = a^{m-n}
• Power of a Power: (a^m)^n = a^{mn}
• Power of a Product: (ab)^n = a^n b^n
• Power of a Quotient: (a/b)^n = a^n / b^n
• Zero Exponent: a^0 = 1 (provided a ≠ 0)
• Negative Exponent: a^{-n} = 1 / a^n

Radicals

• A radical, symbolized by √, denotes the root of a number.
• For example: √x represents the square root of x.

Simplifying Radicals

• Factoring out perfect squares (or cubes) from within the radical.
• For example: √20 = √(4 * 5) = √4 * √5 = 2√5

Rationalizing the Denominator

• Radicals are removed from the denominator of a fraction.
• Achieved by multiplying both the numerator and the denominator by an appropriate expression.
• For example: Multiplying 1/√2 by √2/√2 yields √2/2.

Rational Expressions

• A rational expression is a fraction where the numerator or denominator contains polynomials.

Simplifying Rational Expressions

• Factoring both the numerator and the denominator, then canceling out any common factors.

Operations with Rational Expressions

• Multiplication: Accomplished by multiplying the numerators and denominators directly.
• Division: Achieved by multiplying by the reciprocal of the divisor.
• Addition and Subtraction: Requires finding a common denominator before combining the numerators.

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols.
• Inequality Symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), ≠ (not equal to).

Solving Inequalities

• Similar to solving equations, however, multiplying or dividing by a negative number reverses the inequality sign.

Absolute Value

• The absolute value indicates a number's distance from zero on the number line.
• Denoted as |x|.
• |x| = x if x ≥ 0, and |x| = −x if x < 0.

Solving Absolute Value Equations and Inequalities

• For equations like |x| = a, solve both x = a and x = −a.
• For inequalities like |x| < a, solve −a < x < a, and for |x| > a, solve x < −a or x > a.

Functions

• A function relates inputs to permissible outputs, where each input corresponds to exactly one output.
• Notation: f(x) indicates the output of function f for input x.

Domain and Range

• The domain of a function encompasses all possible input values (x-values).
• The range of a function includes all possible output values (f(x)-values).

Types of Functions

• Linear Functions: f(x) = mx + b
• Quadratic Functions: f(x) = ax^2 + bx + c
• Exponential Functions: f(x) = a^x
• Logarithmic Functions: f(x) = log_b(x)

Graphing

• Equations, inequalities, and functions are visually represented using graphs in algebra.

Coordinate Plane

• The coordinate plane, also known as the Cartesian plane, comprises two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
• Points are represented as ordered pairs (x, y).

Graphing Linear Equations

• Plotting two points that satisfy the equation and drawing a line through them.
• The slope-intercept form (y = mx + b) is used to determine the slope (m) and y-intercept (b).

Graphing Inequalities

• Graphing the boundary line as if it were an equation.
• Solid lines represent ≤ or ≥, while dashed lines represent < or >.
• The region satisfying the inequality is shaded.

Graphing Functions

• Plotting points that satisfy the function and connecting them with a smooth curve.