# Polynomial and Rational Functions

> IB Mathematics: Analysis and Approaches HL · IB AA HL
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-polynomial-and-rational-functions/

This sub-topic covers core properties of polynomials (all degrees) and rational functions, including root-finding, factoring, asymptote identification, and graph sketching, topics that appear regularly across both IB AA HL Paper 1 and Paper 2.

**Prerequisites:** [Quadratic functions and equations](https://www.owlsprep.com/study/ib-math-aa-hl-u1-quadratic-functions/); [Basic function transformations](https://www.owlsprep.com/study/ib-math-aa-hl-u2-function-transformations/)

## Learning objectives

- Identify key properties of polynomial functions of any degree
- Find roots and factor polynomials using core theorems
- Determine asymptotes, holes, and intercepts for rational functions
- Sketch accurate graphs of polynomial and rational functions

## Polynomials: Definitions and Core Theorems

**Polynomial Function** — A function where $n$ is a non-negative integer, $a_i$ are real constants, and $a_n \neq 0$. The leading term is $a_n x^n$, and $n$ is the degree of the polynomial ($\deg(P)$).

*Notation:* P(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0

*Example:* $P(x) = 2x^3 - 5x^2 + 3x - 7$ is a degree 3 polynomial.

**Remainder and Factor Theorems** — When $P(x)$ is divided by $(x - c)$, the remainder equals $P(c)$. If $P(c) = 0$, then $(x - c)$ is a factor of $P(x)$ (this is the Factor Theorem).

**Worked example:** Find the remainder when $P(x) = 3x^4 - 2x^3 + x^2 - 5$ is divided by $(x - 2)$.

1. By the Remainder Theorem, the remainder equals $P(2)$.
2. Substitute $x=2$ into $P(x)$:
3. $$P(2) = 3(2)^4 - 2(2)^3 + (2)^2 - 5$$
4. $$= 3(16) - 2(8) + 4 - 5 = 48 - 16 + 4 - 5 = 31$$
5. The remainder is 31.

> **tip**
>
> Use the Remainder Theorem instead of long division for remainder problems to save time in exams.

## Factoring and Root Finding

By the Fundamental Theorem of Algebra, a degree $n$ polynomial has exactly $n$ roots (counting multiplicities) over the complex numbers. For real polynomials, complex roots occur in conjugate pairs.

**Multiplicity of a Root** — If $(x-c)^k$ is a factor of $P(x)$ but $(x-c)^{k+1}$ is not, then $c$ is a root of multiplicity $k$.

**Worked example:** Factorize $P(x) = x^3 - x^2 - 8x + 12$ completely, given that $x=2$ is a root of multiplicity 2.

1. Since $x=2$ is multiplicity 2, $(x-2)^2$ is a factor. $P(x)$ is degree 3, so the remaining factor is linear: $(ax + b)$.
2. $$P(x) = (x^2 - 4x + 4)(ax + b) = ax^3 + (b - 4a)x^2 + (4a - 4b)x + 4b$$
3. Equate coefficients with the original polynomial:
4. Leading coefficient: $a = 1$, constant term: $4b = 12 \rightarrow b=3$. Check other coefficients: $b - 4a = 3 - 4 = -1$ (matches $x^2$ term), $4a - 4b = 4 - 12 = -8$ (matches $x$ term).
5. Final factorization:
6. $$P(x) = (x - 2)^2(x + 3)$$

**Check your understanding**

1. What is the sum of the roots of $2x^3 - 6x^2 + 4x - 10 = 0$?

   - -3
   - 3
   - -6
   - 6

   *Answer:* 3

   *Why:* By Vieta's formula, sum of roots for $ax^3 + bx^2 + cx + d = 0$ is $-b/a = -(-6)/2 = 3$.

## Rational Functions: Asymptotes and Holes

**Rational Function** — A function written as the ratio of two polynomials $P(x)$ (numerator) and $Q(x)$ (denominator, not the zero polynomial). The domain excludes all roots of $Q(x)$.

*Notation:* R(x) = \frac{P(x)}{Q(x)}

1. **Vertical asymptotes**: Occur at roots of the denominator that are not roots of the numerator. Common roots create *holes*, not asymptotes.
2. **Horizontal asymptotes**: If $\deg(P) < \deg(Q)$, asymptote at $y=0$. If $\deg(P) = \deg(Q)$, asymptote at $y = \frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$. No horizontal asymptote if $\deg(P) > \deg(Q)$.
3. **Oblique (slant) asymptotes**: Occur when $\deg(P) = \deg(Q) + 1$, found via polynomial long division.

**Worked example:** Find all asymptotes of $R(x) = \frac{2x^2 - 2x - 12}{x^2 - 4x + 3}$.

1. Factor numerator and denominator:
2. $$R(x) = \frac{2(x-3)(x+2)}{(x-1)(x-3)}$$
3. The common factor $(x-3)$ means there is a hole at $x=3$, not a vertical asymptote. Simplify to $R(x) = \frac{2(x+2)}{x-1}, x \neq 3$.
4. Vertical asymptote at the root of the simplified denominator: $x=1$.
5. Degrees of numerator and denominator are equal, so horizontal asymptote is $y = 2/1 = 2$.
6. Final result: Vertical asymptote $x=1$, horizontal asymptote $y=2$, hole at $(3, 5)$.

## Sketching Graphs

Graph sketching questions require you to mark all key features: intercepts, roots (with multiplicity behavior), turning points, asymptotes, and end behavior. For polynomials, end behavior is determined by the leading term:

- Even degree: Both ends go the same direction (up if leading coefficient positive, down if negative)
- Odd degree: Ends go opposite directions (right end up if leading coefficient positive, right end down if negative)
- Root behavior: Odd multiplicity = graph crosses x-axis; Even multiplicity = graph touches x-axis and turns

**Worked example:** Sketch the graph of $P(x) = x(x+1)(x-2)^2$.

1. Degree is $1+1+2 = 4$, leading coefficient positive, so both ends of the graph point upwards.
2. Roots: $x=-1$ (multiplicity 1, crosses), $x=0$ (multiplicity 1, crosses), $x=2$ (multiplicity 2, touches).
3. Y-intercept at $P(0) = 0$, so the graph passes through the origin.
4. Check sign of $P(x)$ in each interval: $x<-1$ (positive), $-1<x<0$ (negative), $0<x<2$ (negative), $x>2$ (positive). Connect the points, showing crossing at $-1$ and $0$, touching at $2$, with both ends pointing up.

> **Exam Tip**
>
> Always label all intercepts, roots, and asymptotes on your sketch. Unlabeled key features lose points even if the shape is correct.

> **Exam tip:** Sketching questions are usually 4-6 marks in IB exams, so allocate your time accordingly.

## Common pitfalls

- **Wrong:** Calling a common root of numerator/denominator a vertical asymptote instead of a hole
  - Why it fails: IB examiners explicitly test the distinction between holes and asymptotes
  - Correct: Always factor both polynomials first, cancel common factors before identifying asymptotes, and note the position of any holes
- **Wrong:** Using the constant term instead of the leading coefficient to find horizontal asymptotes when degrees are equal
  - Why it fails: Confusion between intercept formulas and asymptote rules
  - Correct: Remember: horizontal asymptotes depend on leading terms, not constant terms
- **Wrong:** Forgetting that multiplicity changes how the graph behaves at a root
  - Why it fails: Focusing only on the location of the root, not its multiplicity
  - Correct: Memorize: odd multiplicity = cross the x-axis, even multiplicity = touch the x-axis
- **Wrong:** Claiming an asymptote does not exist because the graph crosses it at a finite x-value
  - Why it fails: Misunderstanding that asymptotes describe end behavior, not no intersection
  - Correct: Only vertical asymptotes cannot be crossed; horizontal and oblique asymptotes can be crossed at finite x-values
- **Wrong:** Assuming an odd-degree polynomial has all real roots
  - Why it fails: Misremembering the Fundamental Theorem of Algebra, which only guarantees at least one real root for odd degree
  - Correct: Check possible roots with the Rational Root Theorem or Descartes' Rule of Signs before confirming the number of real roots

## Cheatsheet

| Feature | Polynomial Rule | Rational Function Rule |
| --- | --- | --- |
| Degree | $n = $ highest power of $x$ | No standard degree |
| Vertical Asymptotes | None | Roots of simplified denominator |
| Horizontal Asymptote | None | $\deg(P) < \deg(Q) \rightarrow y=0$; $\deg(P) = \deg(Q) \rightarrow y = \frac{a_n}{b_m}$; else none |
| Oblique Asymptote | None | Exists if $\deg(P) = \deg(Q)+1$ from division |
| Root Behavior | Odd multiplicity: cross x-axis; Even: touch | Same for roots of the numerator |
| End Behavior | Determined by leading term | Determined by leading terms of numerator/denominator |

## What's next

Polynomial and rational functions are foundational for almost all other topics in IB AA HL, from differential and integral calculus to complex numbers and statistical modeling. Mastering root finding, factoring, and graph behavior is critical for solving polynomial equations, integrating rational functions via partial fractions, and analyzing the end behavior of a wide range of functions beyond this sub-topic. IB examiners frequently test combinations of these properties with calculus, so a solid understanding here will save you time and points on exam day.

- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-exponential-and-logarithmic-functions/)
- [Basic trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-basic-trigonometric-functions/)
- [Inverse trigonometric functions (HL only)](https://www.owlsprep.com/study/ib-math-aa-hl-u2-inverse-trigonometric-functions/)

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