College Algebra Overview

Introduction

• Basic introduction to college algebra.
• Overview of exponents, polynomials, linear equations, inequalities, systems of equations, and functions.

Exponents

• Multiplication: When multiplying with the same base, add exponents (e.g., (x^2 \times x^5 = x^{7})).
• Division: When dividing, subtract the exponents (e.g., (x^5 / x^2 = x^3)).
• Negative Exponents: Move the base to the other side of the fraction (e.g., (x^4 / x^7 = 1/x^3)).
• Power of a Power: Multiply exponents (e.g., ((x^3)^4 = x^{12})).
• Zero Exponent: Any base raised to zero is one.

Polynomials

• Simplifying Expressions: Combine like terms (e.g., (5x + 7x = 12x)).
• Distributing: Distribute negative signs when simplifying.
• FOIL Method: Used to multiply two binomials (First, Outer, Inner, Last).

Linear Equations

• Solving Equations: Isolate (x) using inverse operations (e.g., addition for subtraction).
• Graphing: Use slope-intercept form (y = mx + b).

Inequalities

• Solve like equations, but flip the inequality sign when multiplying or dividing by a negative.
• Graph solutions on a number line using open or closed circles.

Absolute Value

• Definition: Non-negative value of a number.
• Equations: Solve by splitting into two cases.
• Inequalities: Similar to equations, but handle direction of inequality carefully.

Quadratic Equations

• Factoring: Look for perfect squares or use difference of squares.
• Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
• Completing the Square: Another method for solving quadratics.

Systems of Equations

• Substitution: Solve one equation for a variable and substitute in the other.
• Elimination: Align equations to cancel out a variable.
• Solution is the intersection of lines in graphical method.

Functions

• Function Notation: (f(x)) notation for functions.
• Composition of Functions: (f(g(x))) and (g(f(x))).
• Inverse Functions: Switch (x) and (y) and solve for new (y).

Graphing Transformations

• Linear Functions: Use y-intercept and slope.
• Quadratic Functions: Vertex form (y = a(x-h)^2 + k).
• Transformations: Shifts, reflections, and stretches.

Conclusion

• Video recap of algebra basics.
• Encouraged to explore more detailed topics in college algebra and other subjects such as physics, chemistry, and calculus.