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Equivalent Representations of Polynomial and Rational Expressions

Alice White

Alice White

7 min read

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Study Guide Overview

This AP Precalculus study guide covers equivalent representations of polynomial and rational expressions (factored and standard forms), quotients of polynomials (polynomial long division), and the Binomial Theorem (Pascal's Triangle). It also reviews high-priority topics such as transformations of functions, trigonometric functions, vectors, matrices, and limits, including common question types and exam tips.

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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:

Key Concept

Easily identify real zeros (x-intercepts) and the domain of the function.

- 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 (β†—).
      • Odd degree with a negative leading coefficient: Starts up, ends down (β†˜).
    • 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).

    A function in its standard form.

    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. βž—

  • 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.

    Function 2x^2+7x-4 divided by x-3; the final reminder is 35.

    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).

    Function x^3+x^2+x+0 divided by x-1. The answer x^2+x+2+2/x-1

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

  • Why it's useful:

    • Finding slant asymptotes of rational functions. πŸ‘πŸΌ

The Binomial Theorem

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

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

    The left column is exponents, the middle column is pascal’s triangle, and the right column is binomial expansion.

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

  • Formula: (a + b)^n = βˆ‘ (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. πŸ€“
    • Simplifying complex expressions and finding zeros.

    Function (2x+3)^5 getting expanded through the binomial expansion theorem

    Caption: Example of binomial expansion.

Memory Aid

Pascal's Triangle Memory Aid: Remember that each number is the sum of the two numbers directly above it. The edges are always 1. Start with 1 at the top.

Memory Aid

Binomial Theorem Memory Aid: Think of it as a combination of Pascal's Triangle coefficients and descending powers of 'a' and ascending powers of 'b'.

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.

Exam Tip

Last-Minute Tips

  • 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.
Exam Tip

FRQ Tip: Always justify your answers. Even if your final answer is correct, you might not get full credit if you don't show your reasoning.

Common Mistake

Common Mistake: Forgetting to consider domain restrictions when working with rational functions or logarithms.

Practice Questions

Practice Question

Multiple Choice Questions

  1. What is the end behavior of the polynomial function f(x)=βˆ’3x5+2x3βˆ’x+7f(x) = -3x^5 + 2x^3 - x + 7? (A) As xβ†’βˆžx \to \infty, f(x)β†’βˆžf(x) \to \infty and as xβ†’βˆ’βˆžx \to -\infty, f(x)β†’βˆ’βˆžf(x) \to -\infty (B) As xβ†’βˆžx \to \infty, f(x)β†’βˆ’βˆžf(x) \to -\infty and as xβ†’βˆ’βˆžx \to -\infty, f(x)β†’βˆžf(x) \to \infty (C) As xβ†’βˆžx \to \infty, f(x)β†’βˆžf(x) \to \infty and as xβ†’βˆ’βˆžx \to -\infty, f(x)β†’βˆžf(x) \to \infty (D) As xβ†’βˆžx \to \infty, f(x)β†’βˆ’βˆžf(x) \to -\infty and as xβ†’βˆ’βˆžx \to -\infty, f(x)β†’βˆ’βˆžf(x) \to -\infty

  2. Which of the following is a possible vertical asymptote of the rational function g(x)=x2βˆ’4x2βˆ’3x+2g(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 x3x^3 term in the expansion of (2xβˆ’1)5(2x - 1)^5? (A) -80 (B) 40 (C) 80 (D) -40

Free Response Question

Consider the rational function h(x)=2x2+5xβˆ’3xβˆ’1h(x) = \frac{2x^2 + 5x - 3}{x - 1}.

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

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

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

(d) Sketch a graph of h(x)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! πŸ’ͺ

Question 1 of 10

πŸŽ‰ What are the x-intercepts of the polynomial function f(x)=(x+1)(xβˆ’4)f(x) = (x+1)(x-4)?

x = 1 and x = -4

x = -1 and x = 4

x = 0 and x = -4

x = 1 and x = 4