College Algebra Overview

Basic Algebraic Operations

Multiplying with Exponents

• Example: (x^2 \times x^5)
  • When multiplying common bases, add exponents: (x^{2+5} = x^7)
  • Explanation: Think of (x^2) as (x \times x) and (x^5) as five (x)'s multiplied together.

Dividing with Exponents

• Example: (x^5 \div x^2)

  • Subtract exponents: (x^{5-2} = x^3)
  • Explanation: Cancel out the common (x)'s.

• Example: (x^4 \div x^7)

  • Subtract to find negative exponent: (x^{4-7} = x^{-3})
  • Explanation: Move to denominator to change sign: (1/x^3).

Exponentiation

• Example: ((x^3)^4)
  • Multiply exponents: (x^{3 \times 4} = x^{12})
  • Explanation: Imagine multiplying four (x^3)'s together results in 12 (x)'s.

Zero Power

• Anything raised to the zero power equals 1.

Simplifying Algebraic Expressions

Combining Like Terms

• Example: (5x + 3 + 7x - 4)
  • Combine like terms: (5x + 7x = 12x)
  • Combine constants: (3 - 4 = -1)

Distributive Property

• Example: (5x^2 - 3x + 7 - (4x^2 + 8x + 11))
  • Distribute negative sign: (5x^2 - 3x + 7 - 4x^2 - 8x - 11)
  • Combine like terms: (x^2 + 5x - 4)

Operations with Polynomials

FOIL Method

• Example: ((3x - 5)(2x - 6))
  • First: (3x \times 2x = 6x^2)
  • Outside: (3x \times -6 = -18x)
  • Inside: (-5 \times 2x = -10x)
  • Last: (-5 \times -6 = 30)
  • Combine: (6x^2 - 28x + 30)

Expanding Squares

• Example: ((2x - 5)^2)
  • Use FOIL: ((2x - 5)(2x - 5) = 4x^2 - 20x + 25)

Solving Linear Equations

Simple Equations

• Example: (x + 6 = 11)
  • Subtract 6: (x = 5)

Solving by Division

• Example: (4x = 8)
  • Divide by 4: (x = 2)

Complex Equations

• Example: (3x + 5 = 26)
  • Subtract 5: (3x = 21)
  • Divide by 3: (x = 7)

Using Distributive Property

• Example: (4(2x - 7) + 8 = 20)
  • Simplify and solve: (2x - 7 = 3), (x = 5)

Graphing and Solving Inequalities

Solving Linear Inequalities

• Example: (2x + 5 > 11)
  • Solve like an equation: (x > 3)

Graphing

• Plot on number line; open circle if not inclusive.

Interval Notation

• Express solutions like ((3, \infty)) or (2, \infty))

Solving with Negative Coefficients

• Example: (-3x \geq 9)
  • Divide and flip sign: (x \leq -3)

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

• Example: (|2x + 3| = 11)
  • Two equations: (2x + 3 = 11) and (2x + 3 = -11)

Solving Absolute Value Inequalities

• Example: (|3x - 1| > 5)
  • Two cases: (3x - 1 > 5) or (3x - 1 < -5)

Graphing Linear Equations

Slope-Intercept Form

• Form: (y = mx + b)
• Example: (y = 2x - 3)
  • Start at (b), slope (m) is rise/run.

Standard Form

• Form: (Ax + By = C)
• Find intercepts to graph.

Function Transformations

Absolute Value and Quadratic Functions

• Transform by shifting, reflection, and stretching.

Quadratic Equations

Solving by Factoring

• Example: (x^2 - 25 = 0)
  • Difference of squares: ((x + 5)(x - 5))

Solving with Quadratic Formula

• Equation: (ax^2 + bx + c = 0)
• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

Systems of Equations

Solving by Elimination

• Align and add/subtract equations to eliminate a variable.

Solving by Substitution

• Substitute expressions to solve for variables.

Functions and Notation

Evaluating Functions

• Replace variables and calculate.

Composite Functions

• Notation: (f(g(x)))
• Substitute entire functions into each other.

Inverse Functions

• Swap x and y, solve for y to find inverse.

Miscellaneous Topics

Imaginary Numbers

• i is the square root of -1.

Solving Systems with Substitution and Graphing

• Substitute and graph to find points of intersection.

These notes offer a concise overview of key algebra topics and techniques, helping to review foundational concepts. For more detailed examples and practice, refer to additional algebra resources or textbooks.