#AP Pre-Calculus: Night Before Review 🚀

Hey! Let's get you prepped and confident for your AP Pre-Calculus exam. This guide is designed to be quick, clear, and super helpful for your last-minute review. Let's do this!

#1.11 Equivalent Representations of Polynomial and Rational Expressions

#Various Forms of Polynomial and Rational Expressions

Understanding how to switch between different forms of polynomial and rational expressions is KEY for the exam. Let's break it down:

#Factored Form

• What it is: A polynomial or rational function expressed as a product of its factors.

• #Why it's useful:

- Reveals **vertical asymptotes** and **holes** (removable discontinuities). - Provides insight into the **range** of the function.

![A function in its factored form.](https://user-images.githubusercontent.com/98941488/288914671-50612083-6e87-4124-8158-77279151899e.gif)

*Caption: A function in its factored form. Notice how the x-intercepts are easily visible.* 

#Standard Form

• What it is: A polynomial written in descending order of powers, or a rational function with polynomials in standard form in numerator and denominator.

• Why it's useful:

  • Determines end behavior of the function. 🧍
  • For polynomials:
    • Even degree with a positive leading coefficient: Ends go up (↑↑).
    • Even degree with a negative leading coefficient: Ends go down (↓↓).
    • Odd degree with a positive leading coefficient: Starts down, ends up (↗).
    • Odd degree with a negative leading coefficient: Starts up, ends down (↘).
  • For rational functions:
    • Numerator degree < denominator degree: Horizontal asymptote at y = 0. - Numerator degree = denominator degree: Horizontal asymptote at the ratio of leading coefficients.
    • Numerator degree > denominator degree: No horizontal asymptote (may have a slant asymptote).
  

  Caption: A function in its standard form. The leading coefficient and degree help determine end behavior.

• Graphing:

  • Visual representation of the function's behavior.
  • Reveals maxima, minima, and points of inflection.
  • Shows shape and symmetry.

#Quotients of Two Polynomial Functions

#Polynomial Long Division

• What it is: A method to divide one polynomial by another, similar to numerical long division. ➗

• How it works:

  1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
  2. Multiply the divisor by this term and subtract from the dividend.
  3. Repeat until the degree of the new dividend is less than the degree of the divisor.
  

  Caption: Example of polynomial long division.

• Result: f(x) = g(x)q(x) + r(x)

  • q(x) is the quotient.
  • r(x) is the remainder (degree is always less than the divisor).
  

  Caption: Another example of polynomial long division. Notice how the remainder is expressed.

• Why it's useful:

  • Finding slant asymptotes of rational functions. 👍🏼

#The Binomial Theorem

• What it is: A method to expand expressions of the form (a + b)^n. 📐

• How it works: Uses Pascal's Triangle to determine coefficients.

  

  Caption: Pascal's Triangle and its relation to binomial expansion.

• Formula: (a + b)^n = ∑ (n choose k) * a^(n-k) * b^k, where (n choose k) is the binomial coefficient.

• Why it's useful:

  • Expanding polynomials of the form p(x) = (x + c)^n quickly. 🤓
  • Simplifying complex expressions and finding zeros.
  

  Caption: Example of binomial expansion.

#Final Exam Focus

#High-Priority Topics

• Transformations of Functions: Shifting, stretching, reflecting graphs.
• Polynomial and Rational Functions: Zeros, asymptotes, end behavior, graphing.
• Trigonometric Functions: Unit circle, identities, graphs, modeling.
• Vectors: Operations, dot product, applications.
• Matrices: Operations, determinants, inverses, solving systems of equations.
• Limits: Basic limit laws, continuity, end behavior of functions.

#Common Question Types

• Multiple Choice: Testing conceptual understanding and quick calculations.
• Free Response: In-depth analysis, multi-step problem solving, explanations.

#

• Time Management: Don't spend too long on one question. Mark it and come back.
• Common Pitfalls:
  • Incorrectly applying transformations.
  • Forgetting to check for extraneous solutions.
  • Misinterpreting the question's context.
• Strategies:
  • Read questions carefully.
  • Show all your work (even if you think it's obvious).
  • Use your calculator effectively, but don't rely on it for everything.
  • Double-check your answers.

#Practice Questions

Practice Question

Multiple Choice Questions

1. What is the end behavior of the polynomial function $f(x) = -3x^5 + 2x^3 - x + 7$? (A) As $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$ (B) As $x \to \infty$, $f(x) \to -\infty$ and as $x \to -\infty$, $f(x) \to \infty$ (C) As $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to \infty$ (D) As $x \to \infty$, $f(x) \to -\infty$ and as $x \to -\infty$, $f(x) \to -\infty$

2. Which of the following is a possible vertical asymptote of the rational function $g(x) = \frac{x^2 - 4}{x^2 - 3x + 2}$? (A) x = -2 (B) x = 1 (C) x = 4 (D) x = 0

3. What is the coefficient of the $x^3$ term in the expansion of $(2x - 1)^5$? (A) -80 (B) 40 (C) 80 (D) -40

Free Response Question

Consider the rational function $h(x) = \frac{2x^2 + 5x - 3}{x - 1}$.

(a) Use polynomial long division to rewrite $h(x)$ in the form $q(x) + \frac{r(x)}{x-1}$. [2 points]

(b) Identify any vertical asymptotes of $h(x)$. [1 point]

(c) Identify any slant asymptotes of $h(x)$. [2 points]

(d) Sketch a graph of $h(x)$, including any asymptotes and intercepts. [4 points]

Scoring Rubric

(a) Polynomial long division: 2 points - 1 point for correct quotient - 1 point for correct remainder

(b) Vertical asymptotes: 1 point - 1 point for identifying x=1

(c) Slant asymptotes: 2 points - 1 point for identifying the equation of the slant asymptote - 1 point for showing the work or reasoning

(d) Graph: 4 points - 1 point for the correct x-intercepts - 1 point for the correct y-intercept - 1 point for the correct asymptotes - 1 point for the correct general shape of the graph

You've got this! Go ace that exam! 💪