Overview
This lecture provides a comprehensive introduction to core topics in College Algebra, covering exponent rules, simplifying expressions, solving equations and inequalities, graphing, factoring, and functions.
Exponent Rules
- When multiplying like bases (e.g., x² * x⁵), add the exponents: x^(2+5) = x⁷.
- When dividing like bases (e.g., x⁵ / x²), subtract the exponents: x^(5-2) = x³.
- Negative exponents indicate reciprocals: x⁻³ = 1/x³
- Raising a power to a power, multiply exponents: (x³)⁴ = x¹².
- Any value raised to the zero power equals one: x⁰ = 1.
Simplifying Expressions & Combining Like Terms
- Combine terms with the same variables and exponents (like terms).
- When subtracting polynomials, distribute negative signs before combining like terms.
Multiplying Polynomials (FOIL)
- Multiply binomials using the FOIL method: First, Outside, Inside, Last.
- Combine like terms after multiplication.
Solving Linear Equations
- To solve x + 6 = 11, subtract 6 from both sides: x = 5.
- For equations like 4x = 8, divide both sides by 4: x = 2.
- If both multiplication and addition exist, isolate terms step by step.
Solving and Graphing Inequalities
- Solve inequalities like equations; if multiplying/dividing by a negative, flip the inequality sign.
- Graph solutions on a number line; open circles for "<" or ">", closed for "≤" or "≥".
- Interval notation: use parentheses ( ) for open and brackets [ ] for closed intervals.
Absolute Value
- The absolute value of any number is always non-negative.
- To solve |x| = a, write two equations: x = a and x = -a.
- For inequalities, split into two cases and solve both.
Compound Inequalities
- For expressions like 1 ≤ 2x + 5 ≤ 13, isolate x in all three parts.
Graphing Linear Equations
- Slope-intercept form: y = mx + b, where m = slope, b = y-intercept.
- Standard form: Ax + By = C; find intercepts by setting x or y to 0.
Graph Transformations
- Adding/subtracting inside affects horizontal shift; outside, vertical shift.
- Negative in front reflects over the axis.
- Absolute value graphs are "V" shaped; quadratics are "U" shaped.
Factoring & Solving Quadratic Equations
- Difference of squares: a² - b² = (a - b)(a + b).
- Factor trinomials by finding two numbers multiplying to c and adding to b.
- Solve by factoring or using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.
Complex Numbers (i)
- i = √(-1), i² = -1, i³ = -i, i⁴ = 1.
Systems of Equations
- Elimination: Add/subtract equations to eliminate variables.
- Substitution: Solve one equation for a variable and substitute.
- Graphing: Solution is the intersection point.
Functions & Function Notation
- Evaluate f(x) by substituting x with given value.
- Composite functions: f(g(x)) means insert g(x) into f.
- Inverse functions: Swap x and y, solve for y; graph is symmetric about y = x.
Key Terms & Definitions
- Exponent — A value showing how many times a base is multiplied by itself.
- Like Terms — Terms with the same variables and exponents.
- Binomial — An algebraic expression with two terms.
- Trinomial — An algebraic expression with three terms.
- Quadratic Equation — Equation of the form ax² + bx + c = 0.
- Absolute Value — The distance a number is from zero, always positive.
- Slope-Intercept Form — Linear equations in the form y = mx + b.
- Standard Form — Linear equations in the form Ax + By = C.
- Composite Function — A function defined by combining two functions, as in f(g(x)).
- Inverse Function — A function that reverses the effect of the original function.
Action Items / Next Steps
- Practice simplifying exponents, combining like terms, and factoring trinomials.
- Solve provided linear, absolute value, and quadratic equations.
- Graph equations using both slope-intercept and standard forms.
- Evaluate and write composite and inverse functions.
- Review quadratic formula and complex numbers.