Module 0d — Polynomial and Rational Functions
Course: MAT1CJ101 — Differential Calculus (prerequisite refresher)
Status: Prerequisite background material — not part of the official 60 taught hours, not examined.
0. Why This Module Exists
Module IV of the main notes (Graphing and Asymptotes) uses calculus — derivatives — to locate turning points and confirm asymptotic behaviour precisely. But you should already be able to sketch a rough shape of a polynomial or rational function from algebra alone: degree, leading coefficient, roots, and simple limits at infinity. This module builds that algebraic groundwork so Module IV can focus on what calculus adds, not on re-teaching factoring.
1. Polynomial Functions
1.1 Definition and Terminology
A polynomial function has the form \(p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0,\) where $n$ is a non-negative integer and $a_n \neq 0$.
- Degree $= n$ (the highest power of $x$ with a nonzero coefficient)
- Leading coefficient $= a_n$
- Leading term $= a_n x^n$
- Constant term $= a_0$
| Degree | Name | General shape |
|---|---|---|
| 0 | Constant | Horizontal line |
| 1 | Linear | Straight line |
| 2 | Quadratic | Parabola |
| 3 | Cubic | S-shaped curve |
| $n$ | Degree-$n$ polynomial | Up to $n - 1$ turning points, up to $n$ real roots |
1.2 End Behaviour
The leading term alone determines what happens as $x \to \pm\infty$ — every other term becomes negligible in comparison for large $\lvert x \rvert$.
| Degree $n$ | Leading coeff $a_n$ | As $x \to \infty$ | As $x \to -\infty$ |
|---|---|---|---|
| Even | positive | $\to +\infty$ | $\to +\infty$ |
| Even | negative | $\to -\infty$ | $\to -\infty$ |
| Odd | positive | $\to +\infty$ | $\to -\infty$ |
| Odd | negative | $\to -\infty$ | $\to +\infty$ |
Worked Example 1: Describe the end behaviour of $p(x) = -2x^5 + 3x^3 - x + 7$.
Leading term: $-2x^5$. Degree 5 is odd, leading coefficient $-2$ is negative.
From the table (odd, negative): $x \to \infty$ gives $p(x) \to -\infty$; $x \to -\infty$ gives $p(x) \to +\infty$.
Sanity check by direct reasoning: for very large $x$, $-2x^5$ dominates all other terms. \(x = 100: \ -2(100)^5 = -2 \times 10^{10}, \text{ overwhelmingly negative — matches "} \to -\infty \text{ as } x\to\infty\text{."}\)
1.3 Factoring Techniques
| Technique | When to use | Example |
|---|---|---|
| Common factor | Every term shares a factor | $3x^3-6x^2 = 3x^2(x-2)$ |
| Difference of squares | $a^2-b^2$ | $x^2-9 = (x-3)(x+3)$ |
| Difference/sum of cubes | $a^3 \pm b^3$ | $x^3-8 = (x-2)(x^2+2x+4)$ |
| Trinomial factoring | $ax^2+bx+c$ | $x^2-x-6 = (x-3)(x+2)$ |
| Grouping | Four-term polynomials | $x^3+2x^2-3x-6 = x^2(x+2)-3(x+2) = (x+2)(x^2-3)$ |
| Rational Root Theorem + synthetic division | Higher-degree polynomials | see Example 3 |
Worked Example 2: Factor $p(x) = x^3 - 2x^2 - 5x + 6$ by grouping after finding one root by inspection.
Try $x = 1$: $p(1) = 1 - 2 - 5 + 6 = 0$. So $(x - 1)$ is a factor.
Divide $p(x)$ by $(x - 1)$ using synthetic division (coefficients $1, -2, -5, 6$):
1 | 1 -2 -5 6
| 1 -1 -6
| 1 -1 -6 0 ← remainder 0, confirms (x-1) is a factor
Quotient: $x^2 - x - 6 = (x-3)(x+2)$
\[p(x) = (x - 1)(x - 3)(x + 2)\]Roots: $x = 1, 3, -2$. Check degree: 3 roots for a degree-3 polynomial. $\checkmark$
1.4 The Rational Root Theorem
For $p(x) = a_n x^n + \cdots + a_0$ with integer coefficients, every rational root $p/q$ (in lowest terms) must have $p$ dividing $a_0$ and $q$ dividing $a_n$. This narrows an infinite search down to a short finite list of candidates.
Worked Example 3: Find all rational roots of $p(x) = 2x^3 - 3x^2 - 11x + 6$.
$a_0 = 6$, divisors: $\pm1, \pm2, \pm3, \pm6$
$a_n = 2$, divisors: $\pm1, \pm2$
Candidates $p/q$: $\pm1, \pm2, \pm3, \pm6, \pm1/2, \pm3/2$
Test $x = 3$: $2(27) - 3(9) - 11(3) + 6 = 54 - 27 - 33 + 6 = 0$. Root found: $x = 3$.
Synthetic division by $(x - 3)$, coefficients $2, -3, -11, 6$:
3 | 2 -3 -11 6
| 6 9 -6
| 2 3 -2 0
Quotient: $2x^2 + 3x - 2 = (2x - 1)(x + 2)$
Full factorization: $p(x) = (x - 3)(2x - 1)(x + 2)$
Roots: $x = 3, x = 1/2, x = -2$
1.5 Multiplicity of Roots and Graph Behaviour
If $(x - r)^k$ is a factor of $p(x)$ with $k$ the largest such power, $r$ is a root of multiplicity $k$.
| Multiplicity | Graph behaviour at $x = r$ |
|---|---|
| Odd ($1, 3, \ldots$) | Graph crosses the $x$-axis |
| Even ($2, 4, \ldots$) | Graph touches the $x$-axis and turns back (doesn’t cross) |
Worked Example 4: Describe the roots and crossing behaviour of $p(x) = (x-1)^2(x+2)^3$.
Root $x = 1$, multiplicity 2 (even) $\to$ graph touches $x$-axis at $x=1$, doesn’t cross
Root $x = -2$, multiplicity 3 (odd) $\to$ graph crosses $x$-axis at $x=-2$
Degree $= 2 + 3 = 5$ (odd), leading coefficient positive (both factors expand with $+1$ leading terms) $\to$ end behaviour: $x\to\infty$ gives $p\to+\infty$, $x\to-\infty$ gives $p\to-\infty$ (odd, positive row).
2. Rational Functions
2.1 Definition
A rational function has the form $r(x) = p(x)/q(x)$, where $p$ and $q$ are polynomials and $q$ is not the zero polynomial. Its natural domain excludes every $x$ where $q(x) = 0$.
2.2 Vertical Asymptotes and Holes
Find the zeros of the denominator. For each zero $x = a$:
- If $(x - a)$ is not also a factor of the numerator: vertical asymptote at $x = a$.
- If $(x - a)$ is also a factor of the numerator (and cancels): usually a hole (removable discontinuity) at $x = a$ instead of an asymptote — provided it doesn’t remain in the denominator with higher multiplicity than in the numerator.
Worked Example 5: Find the vertical asymptotes/holes of $r(x) = (x^2 - 1)/(x^2 - 3x + 2)$.
Factor numerator: $x^2 - 1 = (x-1)(x+1)$
Factor denominator: $x^2 - 3x + 2 = (x-1)(x-2)$
Common factor $(x-1)$ cancels (for $x \neq 1$): $r(x) = \dfrac{x+1}{x-2}, \ x \neq 1$
$x = 1$: cancelled factor $\to$ HOLE at $x = 1$ (not an asymptote). Hole’s $y$-coordinate: plug $x=1$ into the simplified form $\dfrac{x+1}{x-2} = \dfrac{2}{-1} = -2$. Hole at $(1, -2)$
$x = 2$: remains in the denominator of the simplified form, not in the numerator $\to$ VERTICAL ASYMPTOTE at $x = 2$
2.3 Horizontal Asymptotes (Algebraic Preview)
Compare the degree of the numerator ($\deg p = m$) to the degree of the denominator ($\deg q = n$):
| Comparison | Horizontal asymptote |
|---|---|
| $m < n$ | $y = 0$ |
| $m = n$ | $y = $ (leading coeff of $p$)/(leading coeff of $q$) |
| $m > n$ | none (function grows without bound; may have a slant asymptote if $m = n+1$) |
This table is an algebraic shortcut for limits at infinity — Module I proves it rigorously using limit laws; here we just state the pattern so you can sketch a rough shape early.
Worked Example 6: Find the horizontal asymptote of $r(x) = (3x^2 + x)/(2x^2 - 5)$.
$\deg(\text{numerator}) = 2 = \deg(\text{denominator})$. Equal degree case.
\[\text{Horizontal asymptote: } y = \frac{\text{leading coeff of numerator}}{\text{leading coeff of denominator}} = \frac{3}{2}\]Sanity check with a large $x$, say $x = 1000$: numerator $\approx 3(1000)^2 = 3{,}000{,}000$, denominator $\approx 2(1000)^2 = 2{,}000{,}000$, ratio $\approx 1.5 = 3/2$ $\checkmark$ matches
Worked Example 7: Find the horizontal asymptote of $r(x) = (x + 4)/(x^2 - 1)$.
$\deg(\text{numerator}) = 1 < \deg(\text{denominator}) = 2$.
Horizontal asymptote: $y = 0$
Sanity check, $x = 1000$: $1004/999999 \approx 0.001$, very close to 0. $\checkmark$
2.4 Putting It Together — Sketching a Rough Shape
Worked Example 8: Sketch the key features of $r(x) = (x - 3)/(x^2 - 4)$.
Domain: $x^2 - 4 = 0 \implies x = \pm2$. Domain: $\mathbb{R} \setminus {-2, 2}$
Numerator zero: $x = 3$ $\to$ $x$-intercept at $(3, 0)$
$y$-intercept: $r(0) = (0-3)/(0-4) = -3/-4 = 3/4$ $\to$ $(0, 3/4)$
Vertical asymptotes: denominator zero at $x=\pm2$; numerator $(x-3)$ does NOT vanish at $\pm2$, so BOTH are genuine vertical asymptotes (no cancellation). $x = -2$ and $x = 2$
Horizontal asymptote: $\deg(\text{num})=1 < \deg(\text{denom})=2$ $\to$ $y = 0$
Sign chart (critical points $-2, 2, 3$ split the line into 4 intervals) to know which side of each asymptote the curve approaches from:
\(x < -2: \ \text{test } x=-3: \ \frac{-6}{5} = -1.2 \ \to \text{ negative}\) \(-2<x<2: \ \text{test } x=0: \ \frac{-3}{-4} = 0.75 \ \to \text{ positive}\) \(2<x<3: \ \text{test } x=2.5: \ \frac{-0.5}{2.25} \approx -0.22 \ \to \text{ negative}\) \(x > 3: \ \text{test } x=4: \ \frac{1}{12} \approx 0.083 \ \to \text{ positive}\)
This sign pattern plus the two vertical asymptotes and one horizontal asymptote is enough to sketch the overall shape without calculus.
3. Summary
| Concept | Key Idea | Worked Example |
|---|---|---|
| Degree & end behaviour | Leading term controls $x\to\pm\infty$ | $-2x^5+\ldots \to -\infty$ as $x\to\infty$ |
| Factoring by grouping | Find one root, synthetic-divide, factor quotient | $x^3-2x^2-5x+6 = (x-1)(x-3)(x+2)$ |
| Rational Root Theorem | Candidates are $\pm$(factor of $a_0$)/(factor of $a_n$) | $2x^3-3x^2-11x+6$ has root $x=3$ |
| Multiplicity | Odd $\to$ crosses axis; even $\to$ touches and turns | $(x-1)^2(x+2)^3$ |
| Vertical asymptote vs. hole | Uncancelled factor $\to$ asymptote; cancelled $\to$ hole | $(x^2-1)/(x^2-3x+2)$: hole at 1, asymptote at 2 |
| Horizontal asymptote | Compare degrees of numerator/denominator | deg equal $\to$ ratio of leading coeffs |
4. Practice Problems
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Describe the end behaviour (as $x\to\infty$ and $x\to-\infty$) of $p(x) = 4x^6 - 3x^2 + 1$.
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Describe the end behaviour of $p(x) = -x^7 + 2x^4 - x$.
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Factor $p(x) = x^3 - 7x + 6$ completely (find one root by inspection, then synthetic divide).
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Use the Rational Root Theorem to list all candidate rational roots of $p(x) = 3x^3 + 2x^2 - 7x + 2$, then find all actual rational roots and factor completely.
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Factor $x^3 + 3x^2 - 4x - 12$ by grouping.
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For $p(x) = (x+1)^3(x-4)^2$, state each root’s multiplicity and whether the graph crosses or touches the $x$-axis there. What is the degree of $p$, and what is its end behaviour (assume positive leading coefficient)?
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Find the domain, and classify each excluded point as a vertical asymptote or a hole, for $r(x) = (x^2 - 4)/(x^2 - x - 6)$.
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Find the domain and vertical asymptotes of $r(x) = (x + 5)/(x^2 + 3x - 10)$.
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Find the horizontal asymptote of $r(x) = (5x^3 - x)/(2x^3 + 4)$.
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Find the horizontal asymptote of $r(x) = (2x + 7)/(x^2 - 9)$.
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Find the horizontal asymptote of $r(x) = (x^3 - 1)/(x + 2)$, or explain why there is none (what happens instead, given the degree comparison?).
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For $r(x) = (x - 1)/(x^2 - 1)$, identify the hole (with its $y$-coordinate) and the vertical asymptote, then state the horizontal asymptote.
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Sketch the key features (domain, intercepts, vertical/horizontal asymptotes, and sign on each interval) of $r(x) = (2x)/(x^2 - 1)$, following the method of Worked Example 8.
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(Harder) A degree-4 polynomial has roots at $x = -1$ (multiplicity 1), $x = 2$ (multiplicity 3), a positive leading coefficient, and no other real roots. Sketch its end behaviour and crossing/touching pattern, and write one possible formula for it.
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(Harder) Explain, using the degree-comparison table, why a rational function can never have both a horizontal asymptote and a slant asymptote at the same time.