Unit 1A Review: Polynomial and Rational Functions

This review covers key concepts of polynomial and rational functions, including identifying zeros, analyzing end behaviors, and understanding symmetry.

Examples

1. Linear Polynomial: f(x) = 2x + 3. Zero at x = -3/2.
2. Quadratic Polynomial: f(x) = x^2 - 4. Zeros at x = -2, 2.
3. Cubic Polynomial: f(x) = x^3 - 27. Zero at x = 3.
4. Quartic Polynomial: f(x) = x^4 - 16. Zeros at x = ±2, ±2i.
5. Quintic Polynomial: f(x) = x^5 - 32x. Zeros at x = 0, ±2√2, ±i2√2.
6. Rational Function: f(x) = (x^2 - 9)/(x + 3). Hole at x = -3.
7. Even Polynomial: f(x) = x^4 + 4x^2. Symmetric about the y-axis.
8. Odd Polynomial: f(x) = x^3 - x. Symmetric about the origin.
9. End Behavior: f(x) = 3x^5. As x → ±∞, f(x) → ±∞.
10. Complex Zeros: f(x) = x^2 + 1. Zeros at x = ±i.
11. Polynomial Division: Divide x^3 + 2x^2 - 5x - 6 by x - 1.
12. Synthetic Division: Use synthetic division to divide x^3 - 4x^2 + 5x - 2 by x - 2.
13. Factoring Polynomials: Factor x^3 - 6x^2 + 11x - 6.
14. Polynomial Graphs: Sketch the graph of f(x) = (x - 1)^2(x + 2).
15. Rational Function Asymptotes: Identify vertical and horizontal asymptotes of f(x) = (2x^2 - 8)/(x^2 - 4).
16. Transformations: Describe transformations of f(x) = x^2 to g(x) = -2(x + 3)^2 + 1.
17. Descartes' Rule of Signs: Use Descartes' rule on f(x) = x^4 - 3x^3 + 2x^2 + x - 5.
18. Intermediate Value Theorem: Apply the Intermediate Value Theorem to f(x) = x^3 - x - 1 on the interval [0, 2].
19. Remainder Theorem: Use the Remainder Theorem to evaluate f(2) for f(x) = x^