Algebra Fundamentals

Algebra

• Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and express relationships.

• Key Concepts:

  • Variables: Symbols (often letters) that represent numbers or quantities. Examples: x, y, z.
  • Constants: Fixed values that do not change. Examples: 5, -3, π.
  • Expressions: Combinations of variables, constants, and operations (e.g., 2x + 3).
  • Equations: Mathematical statements that assert the equality of two expressions (e.g., 2x + 3 = 7).

• Operations:

  • Addition and Subtraction: Basic operations used to combine or remove quantities.
  • Multiplication and Division: Used to scale quantities and divide them into parts.

• Properties:

  • Commutative Property: a + b = b + a or ab = ba
  • Associative Property: (a + b) + c = a + (b + c) or (ab)c = a(bc)
  • Distributive Property: a(b + c) = ab + ac

• Solving Equations:

  • Isolate the variable: Rearranging the equation to get the variable on one side.
  • Inverse operations: Using opposite operations to solve for the variable.
  • Balancing: Applying the same operation to both sides of the equation.

• Types of Equations:

  • Linear Equations: Equations of the form ax + b = c, where the graph is a straight line.
  • Quadratic Equations: Equations of the form ax² + bx + c = 0, which can be solved via factoring, completing the square, or the quadratic formula.
  • Polynomial Equations: Expressions that involve terms with variables raised to whole number powers.

• Functions:

  • Definition: A relation that assigns exactly one output for each input.
  • Notation: f(x) denotes a function of x.
  • Types: Linear, quadratic, polynomial, exponential, etc.

• Graphing:

  • Coordinate System: A plane with an x-axis (horizontal) and a y-axis (vertical).
  • Plotting Points: Points represented as (x, y) in the coordinate plane.
  • Slope and Intercept: Slope (m) represents the steepness of the line; intercept (b) is where the line crosses the y-axis.

• Factoring:

  • Definition: Breaking down an expression into simpler components (factors) which when multiplied give the original expression.
  • Common Techniques:
    • Finding the greatest common factor (GCF).
    • Quadratic factoring (e.g., a² - b² = (a + b)(a - b)).

• Applications:

  • Used in solving real-world problems such as calculating profits, determining dimensions, and optimization.

Definition of Algebra

• Algebra is a branch of math that uses symbols to manipulate and solve equations and represent relationships.

Key Concepts

• Variables: Symbols representing unknown numerical values (e.g., x, y, z).
• Constants: Fixed numerical values that remain unchanged (e.g., 5, -3, π).
• Expressions: Combinations of variables, constants, and mathematical operations (e.g., 2x + 3).
• Equations: Mathematical statements indicating the equality of two expressions (e.g., 2x + 3 = 7).

Mathematical Operations in Algebra

• Addition and Subtraction: Operations used for combining or removing quantities.
• Multiplication and Division: Operations for scaling quantities and dividing them into parts.

Properties of Mathematical Operations

• Commutative Property: Order of operation doesn't affect the outcome (e.g., a + b = b + a , ab = ba).
• Associative Property: Grouping of operations doesn't change the outcome (e.g., (a + b) + c = a + (b + c), (ab)c = a(bc)).
• Distributive Property: Multiplying a sum by a value is equal to multiplying each term of the sum individually (e.g., a(b + c) = ab + ac).

Solving Equations

• Isolating the Variable: Rearranging the equation to have the variable on one side.
• Inverse Operations: Using opposite mathematical operations to solve for the variable.
• Balancing: Applying the same operation to both sides of the equation to maintain equality.

### Types of Equations

• Linear Equations: Equations of the form ax + b = c, where the graph is a straight line.

• Quadratic Equations: Equations of the form ax² + bx + c = 0, which can be solved through factoring, completing the square, or using the quadratic formula.

• Polynomial Equations: Expressions involving terms with variables raised to whole number powers.

  ### Functions in Algebra

• Definition: A relationship assigning a single output for each input.

• Notation: f(x) denotes a function of x.

• Types of Functions: Linear, quadratic, polynomial, exponential, and others.

Graphing Functions

• Coordinate System: A plane with a horizontal x-axis and a vertical y-axis.
• Plotting Points: Points represented as ordered pairs (x, y) on the coordinate plane.
• Slope: Represents the steepness of a line (m).
• Intercept: The point where the line crosses the y-axis (b).

Important Concept: Factoring

• Definition: Breaking down an expression into its simpler multiplicative components.
• Common Techniques:
  • Finding the greatest common factor (GCF).
  • Quadratic factoring (e.g., a² - b² = (a + b)(a - b)).

Applications of Algebra

• Real-world problems: Solving for profits, determining dimensions, and optimization.

### Exponents

• Exponents indicate repeated multiplication of a base number.
• Base is the number being multiplied.
• Exponent indicates the number of times the base is multiplied by itself.
• Example: ( 3^4 ) is the same as ( 3\times3\times3\times3 ), where 3 is the base and 4 is the exponent.

Powers

• Powers are the result of raising a base to an exponent.
• Example: (3^4=81) is the power.

Squares

• Squaring a number is raising it to the power of 2.
• Example: ( 5^2=25 )

Cubes

• Cubing a number is raising it to the power of 3.
• Example: (4^3=64)

Zero Exponent Rule

• Any non-zero base raised to the power of zero equals 1.
• Example: ( 7^0 = 1 )

Negative Exponent Rule

• A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
• Example: ( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} )

Fractional Exponent Rule

• Fractional exponents represent roots.
• The numerator of the fraction indicates the power.
• The denominator of the fraction indicates the root.
• Example: ( 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 )

Laws of Exponents

• Product of Powers: ( a^m \times a^n = a^{m+n} )
• Quotient of Powers: ( \frac{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: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} )

Scientific Notation

• A way to express very large or very small numbers compactly.
• Format: ( a \times 10^n ), where ( 1 \leq a < 10 ) and ( n ) is an integer.
• Example: ( 3.5 \times 10^5 ), represents a number that's 350,000 in standard form.