Precalculus: We consider a proof by induction with divisibility. Show for all n gt 1 that 3 divides 5^n-2^n. The bookkeeping i simplified by using the definition of divisible. We also show where this class of equations come from.

1. Example of Arithmetic Sequence
2. Example of Arithmetic Sum
3. Geometric Proof of Integer Sum Formula
4. Example of Geometric Sum
5. Example of Proof By Induction
6. Example of Proof by Induction 2: 3 divides 5^n - 2^n
7. Example of Proof by Induction 3: n! less than n^n
8. Example of Binomial Theorem
9. Example of a Piecewise-Defined Function
10. Domain of sqrt(x+1) - 1/sqrt(9-x^2)
11. Example of Composition of Functions
12. Graph of y = -2sqrt(x+4) - 3
13. Example of Projectile Motion
14. Inverse Function for f(x) = (x-2)/(x+2)
15. Using Synthetic Division to Evaluate a Polynomial
16. Fast Factorization of Polynomial
17. Multiplying Complex Numbers
18. Complex Conjugate Roots of a Real Polynomial
19. Factoring 3x^4-2x^3+3x-2 Over The Complex Numbers
20. Sketch of Rational Function
21. Partial Fraction Expansion of (x^4+x^3+1)/(x(x^2+1)^2)
22. Evaluation of 3ln(2e^2) + 2ln(3/e) - ln(72e)
23. Domain of ln((x+1)/(x-1)) +ln(x-1)-ln(x+1)
24. Solving an Exponential Equation
25. Memorizing Trig Values: Three Easy Pieces
26. Trig Values for Multiples of pi/5
27. Examples of Values of Trig Functions
28. Calculating Trig Values
29. Examples of Inverse Trig Evaluations
30. Graph of y = -4sin(3x-pi)
31. Graph of y = 2 sec(pi x - pi/3)
32. Example of Converting to Polar Coordinates
33. Example of Graph of Polar Equation
34. Example of Proof of Trig Identity
35. Example of Double Angle Formulas
36. Example of Half Angle Formula
37. Example of Trig Equation
38. Trig Values for Multiples of pi/5
39. Example of De Moivre's Theorem
40. Cube Roots of -8
41. Complex Solutions of z^4-4z^2+16=0
42. Example of Law of Sines
43. Example of Law of Cosines

In this mini-lecture series, Robert Donley (MathDoctorBob) walks you through important advanced topics in PreCalculus through example problems, showing you both the theory behind the problems, and how to solve them. He shows you trigonometric identities and laws such as the Law of Sines and Law of Cosines, complex numbers including cube roots and De Moivre's theorem, arithmetic sequences and sums, geometric sum, proof by induction, the binomial theorem, composition of functions, projectile motion, factorization of polynomials, partial fraction, exponential and inverse functions, and much more.

Whether you are a high school student learning this material for the first time, or someone in need of a quick college algebra refresher, this short course will be invaluable to you.