AP · Polynomial and Rational Functions · 16 min read · Updated 2026-05-10

Polynomial and Rational Functions — AP Precalculus Unit Study Guide

For: AP Precalculus candidates sitting AP Precalculus.

Covers: All 13 subtopics of AP Precalculus Unit 1, including function behavior, rates of change, polynomial zeros and end behavior, rational function features, equivalent expressions, transformations, and function modeling.

You should already know: Basic function notation and domain/range identification, algebraic manipulation of polynomial expressions, introductory limit concepts for end behavior analysis.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Precalculus style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. Why This Unit Matters

This is the first core unit of AP Precalculus, and carries a 30–40% exam weighting, meaning roughly one-third of all points on the AP exam come from content in this unit. It appears in every section of the exam, from simple multiple-choice questions testing asymptote identification to multi-part free-response questions asking you to build and interpret rational models for real-world contexts. This unit formalizes the core function framework that unites all of AP Precalculus: every subsequent unit (exponential, logarithmic, trigonometric, parametric) follows the same pattern you’ll practice here: analyzing behavior, calculating rates of change, finding key features, and building models. Unlike introductory algebra, which focuses mostly on algebraic manipulation, this unit emphasizes connecting algebraic form to graphical behavior and real-world meaning, skills that are explicitly tested on every AP Precalculus exam. Mastery here removes the biggest stumbling blocks for the rest of the course, as all later units repeat the same analytical structure applied to new function types.

2. Unit Concept Map

The unit builds incrementally from general function concepts to specialized analysis and finally application, with all topics relying on earlier foundational skills:

Foundational layer: Change in tandem (function behavior) and Rates of change (average and over equal intervals) establish the core analytical habits you’ll use for every function type: how to describe how output changes as input changes, and how to quantify that change numerically.
Polynomial-specific layer: Polynomial functions and rates of change, Polynomial functions and complex zeros, and Polynomial functions and end behavior apply the foundation to the simpler family of polynomials, starting from the constant nth difference property all the way to factoring and long-run behavior.
Rational-specific layer: All rational function topics (Rational functions and end behavior, Rational functions and zeros, Rational functions and vertical asymptotes, Rational functions and holes) rely directly on your ability to factor and analyze polynomials to find key features of ratios of polynomials.
Unifying algebraic layer: Equivalent representations of polynomial and rational expressions ties the previous sections together by showing how rewriting expressions reveals hidden key features, and Transformations of functions connects this unit’s function families to parent function transformation rules.
Applied layer: Function model selection and assumption articulation and Function model construction and application let you put all prior skills to work solving real-world problems, the highest-complexity skill tested on the exam.

3. A Guided Tour of a Unit-Style Problem

We work through a common multi-skill exam problem to show how multiple central subtopics connect in sequence:

Problem: For $f (x) = \frac{x ^{3} - 4 x ^{2} + x + 6}{x ^{2} - x - 2}$ , identify all holes, vertical asymptotes, zeros, and end behavior.

Step 1: First, use Equivalent representations of polynomial and rational expressions: factor both numerator and denominator to reveal hidden structure. By the Rational Root Theorem, the numerator factors to $(x + 1) (x - 2) (x - 3)$ , and the denominator factors to $(x + 1) (x - 2)$ . This lets us rewrite the function as: $f (x) = x - 3 for all x \neq = - 1, x \neq = 2$

Step 2: Next, apply Rational functions and holes: any input that makes the original denominator zero and cancels out in simplification is a hole, not an asymptote. We find the y-coordinate of each hole by plugging the x-value into the simplified function: hole at $(- 1, - 4)$ and hole at $(2, - 1)$ .

Step 3: Next, check for vertical asymptotes (from Rational functions and vertical asymptotes): after canceling, no factors remain in the denominator, so there are no vertical asymptotes.

Step 4: Find the zero using Rational functions and zeros: the only zero of the simplified function is at $x = 3$ , which is not a hole, so the function has a zero at $(3, 0)$ .

Step 5: Finally, apply Rational functions and end behavior: since the degree of the numerator is exactly 1 greater than the degree of the denominator, the end behavior follows the slant asymptote equal to the simplified function $y = x - 3$ , so as $x \to \pm \infty$ , $f (x)$ approaches $x - 3$ .

This sequence shows how every step relies on mastering earlier subtopics to get the full set of correct key features.

Exam tip for the unit: Always work through problems from factoring/simplification first, before identifying any key features. 90% of errors on multi-step feature problems come from trying to identify features before simplifying.

4. Cross-Cutting Common Pitfalls

These are the most frequent unit-wide mistakes that span multiple subtopics, with specific corrections:

Wrong move: When simplifying a rational function, you cancel common factors before noting that the original denominator cannot equal zero, so you forget to mark holes. Why: Students prioritize getting a simplified form and ignore that the domain is defined by the original function, not the simplified version. Correct move: Always list all x-values that make the original denominator zero before canceling any common factors, then mark any that cancel as holes.
Wrong move: When describing polynomial end behavior, you only use the sign of the leading coefficient and forget to check if the degree is even or odd. Why: Students memorize "positive leading coefficient goes up on the right" and forget left end behavior flips based on degree parity. Correct move: Always note both degree parity and leading coefficient sign before writing end behavior conclusions.
Wrong move: When calculating average rate of change over $[a, b]$ , you reverse the order of subtraction, getting the wrong sign. Why: Students do not anchor the interval to left-to-right order, and randomly subtract. Correct move: Always write average rate of change as $\frac{f ( right endpoint ) - f ( left endpoint )}{right endpoint - left endpoint}$ to get the correct sign and magnitude every time.
Wrong move: After finding all real zeros of a polynomial, you stop counting and do not account for non-real complex zeros. Why: Students associate zeros with x-intercepts on the real graph, so they forget non-real zeros do not appear graphically. Correct move: After factoring, always confirm the total number of zeros (counting multiplicity, real and complex) equals the degree of the polynomial, per the Fundamental Theorem of Algebra.
Wrong move: When building a model for a real-world context, you use the full mathematical domain of the function and do not restrict it to contextually valid values. Why: Students focus on finding the algebraic expression and forget that negative inputs (like negative radius or time) do not make sense in context. Correct move: After constructing any model, always add an explicit domain restriction that matches the scenario’s physical constraints.

5. Quick Check: When To Use Which Subtopic

Test your understanding by matching each question to the correct subtopic, answers are at the end:

Question: What is the average velocity of a falling object between $t = 1$ and $t = 3$ seconds, if height is given by a polynomial?
Question: A 5th-degree polynomial with real coefficients crosses the x-axis exactly 3 times. How many non-real complex zeros does it have?
Question: The graph of a rational function has a break that is not a vertical asymptote at what x-value?
Question: How does the graph of $2 (x + 1)^{3} - 4$ compare to the parent graph of $y = x^{3}$ ?
Question: What value does $f (x) = \frac{3 x ^{2} - 2 x + 1}{7 x ^{2} + 5}$ approach as $x$ becomes very large?

Answers:

Rates of change (average and over equal intervals)
Polynomial functions and complex zeros
Rational functions and holes
Transformations of functions
Rational functions and end behavior

6. Practice Questions (AP Precalculus Unit Style)

Question 1 (Multiple Choice)

Which of the following correctly lists all holes and vertical asymptotes of the function $f (x) = \frac{( x ^{2} - 9 ) ( x - 4 )}{( x + 3 ) ( x ^{2} - 5 x + 4 )}$ ? A: Hole at $x = - 3$ ; vertical asymptote at $x = 1$ B: Hole at $x = - 3$ ; vertical asymptotes at $x = 1$ and $x = 4$ C: Holes at $x = - 3$ and $x = 4$ ; vertical asymptote at $x = 1$ D: Hole at $x = 4$ ; vertical asymptotes at $x = - 3$ and $x = 1$

Worked Solution: First, fully factor all terms to identify common factors and denominator zeros: $x^{2} - 9 = (x - 3) (x + 3)$ , and $x^{2} - 5 x + 4 = (x - 1) (x - 4)$ . Rewriting the function gives $f (x) = \frac{( x - 3 ) ( x + 3 ) ( x - 4 )}{( x + 3 ) ( x - 1 ) ( x - 4 )}$ . The original denominator has zeros at $x = - 3$ , $x = 1$ , and $x = 4$ . Common factors that appear in both numerator and denominator are $(x + 3)$ and $(x - 4)$ , so these correspond to holes. The only remaining zero in the simplified denominator is $x = 1$ , which is a vertical asymptote. This matches option C. Correct answer: C.

Question 2 (Free Response)

Let $f (x) = x^{4} - 5 x^{3} + 5 x^{2} + 5 x - 6$ . (a) Given that $x = 1$ is a zero of $f (x)$ , find all other real and complex zeros of $f (x)$ . (b) Describe the end behavior of $f (x)$ as $x \to \infty$ and as $x \to - \infty$ . (c) If $g (x) = f (x - 1) + 3$ , what is the end behavior of $g (x)$ ?

Worked Solution: (a) Use synthetic division to divide $f (x)$ by $(x - 1)$ , giving the cubic quotient $x^{3} - 4 x^{2} + x + 6$ . Test the next rational root $x = 2$ and divide again, giving the quadratic quotient $x^{2} - 2 x - 3$ , which factors to $(x - 3) (x + 1)$ . All roots are real, so the full set of additional zeros is ${- 1, 2, 3}$ , with no non-real complex zeros. (b) $f (x)$ is degree 4 (even) with a leading coefficient of 1 (positive). For even-degree polynomials with positive leading coefficient, $lim_{x \to \infty} f (x) = \infty$ and $lim_{x \to - \infty} f (x) = \infty$ . (c) Horizontal and vertical shifts do not change the degree or leading coefficient of a polynomial, so end behavior is unchanged: $lim_{x \to \infty} g (x) = \infty$ and $lim_{x \to - \infty} g (x) = \infty$ .

Question 3 (Application / Real-World Style)

A cylindrical produce can is being designed to hold a fixed volume of 1000 cm³. The material for the side (lateral surface) costs 1 cent per square centimeter, and the material for the top and bottom circular bases costs 2 cents per square centimeter. Write a rational function that gives the total production cost of the can as a function of the radius $r$ of the base, then state the domain of the function in context.

Worked Solution: The volume of a cylinder is $V = π r^{2} h = 1000$ , so solve for height to get $h = \frac{1000}{π r ^{2}}$ . The lateral surface area is $2 π r h = 2 π r (\frac{1000}{π r ^{2}}) = \frac{2000}{r}$ , so the cost of the side is $1 \cdot \frac{2000}{r} = \frac{2000}{r}$ cents. The combined area of the two bases is $2 π r^{2}$ , so the cost of the bases is $2 \cdot 2 π r^{2} = 4 π r^{2}$ cents. The total cost function is $C (r) = 4 π r^{2} + \frac{2000}{r}$ . In context, radius must be a positive real number, so the domain is $r > 0$ , or $(0, \infty)$ in interval notation. Interpretation: This function allows manufacturers to calculate the total material cost for any chosen radius, and can be used to find the radius that gives the minimum production cost for a can holding the required volume.

7. Quick Reference Cheatsheet

Category	Formula / Rule	Notes
Average Rate of Change	$\frac{f ( b ) - f ( a )}{b - a}$	Order right endpoint minus left endpoint for correct sign
nth Degree Polynomial nth Difference	Constant, equal to $n! \cdot a_{n}$	$a_{n}$ is leading coefficient, used to confirm polynomial degree
Fundamental Theorem of Algebra	nth degree polynomial has exactly n zeros (counting multiplicity)	Non-real complex zeros come in conjugate pairs for real-coefficient polynomials
Polynomial End Behavior	Even degree, + leading coeff: $lim_{x \to \pm \infty} f (x) = \infty$ ; Even -, $lim_{x \to \pm \infty} f (x) = - \infty$ ; Odd +: $lim_{x \to \infty} f (x) = \infty, lim_{x \to - \infty} f (x) = - \infty$	Only depends on degree parity and leading coefficient sign
Rational Function Hole	$x = a$ is a hole if $(x - a)$ is a common factor of numerator/denominator	Find y-coordinate by plugging $a$ into the simplified function
Rational Vertical Asymptote	$x = a$ is a vertical asymptote if $(x - a)$ is a factor of the simplified denominator	Does not cancel with any numerator factor
Rational End Behavior	deg(num) < deg(den): $y = 0$ ; deg(num)=deg(den): $y = \frac{leading num coeff}{leading den coeff}$ ; deg(num) = deg(den)+1: slant asymptote = division quotient	Describes long-run behavior as $x \to \pm \infty$
Horizontal Shift	$f (x - h)$	Shifts parent graph $h$ units right (left if $h$ is negative)
Vertical Shift	$f (x) + k$	Shifts parent graph $k$ units up (down if $k$ is negative)

8. What's Next

This unit establishes the core function analysis framework that you will reuse for every function family in the rest of AP Precalculus. After completing all subtopics in this unit, you will move on to Unit 2: Exponential and Logarithmic Functions, where you will apply the same sequence of skills: analyzing change in tandem, calculating rates of change, finding key features, and building models to a new function family. Without mastering the core habits (factoring to find key features, connecting algebraic form to graphical behavior, articulating modeling assumptions) you learn here, you will struggle to adapt these skills to new function types later in the course. This unit also builds the average rate of change skills that are the foundation for any future calculus study.