MGU Mathematics Series — Algebra II
Part II: Quadratics, Complex Numbers, and Polynomial Functions
Degree · End Behavior · Zeros · Multiplicity · Division · Theorems
Chapter 5 of 14
MGU Mathematics Series  ·  Algebra II  ·  Library Textbook Edition

Chapter 5:
Polynomial
Functions

Structure, end behavior, zeros, multiplicity, turning points, long division, synthetic division, and the Remainder and Factor Theorems — the complete architecture of polynomial functions

■ Math Level: Algebra II ■

■ 14 Sections  •  Part II of 5 ■

Follows Chapter 4: Complex Numbers  ·  Prepares students for Chapter 6: Polynomial Equations and Theorems

Chapter 5 — Front Matter

Purpose and Overview

Chapter 5 extends your understanding of functions beyond linear and quadratic relationships. You have already studied lines, which grow or shrink at a constant rate, and parabolas, which curve in a predictable U-shape. Polynomial functions generalize both of these. They can curve, cross, bounce, change direction multiple times, and model complex real-world patterns that linear and quadratic functions cannot capture.

This chapter studies polynomial functions through the lens of structure. Every feature of a polynomial's graph — how it rises or falls at the edges, where it crosses or touches the x-axis, how many times it changes direction — is connected to the algebraic form of the function. Learning to read that structure is the central skill of this chapter.

By the end of Chapter 5, you should be able to identify polynomial functions, analyze their degree and leading coefficient, describe end behavior, find zeros and interpret their multiplicity, count turning points, graph polynomial functions by hand, divide polynomials using long division and synthetic division, and apply the Remainder and Factor Theorems. These skills form the foundation for Chapter 6.

Opening Question

A roller coaster designer is building a new ride. The height of the track, in feet above the ground, is modeled by the function $h(x) = -0.002x^4 + 0.08x^3 - 0.9x^2 + 4x + 10$, where $x$ is the horizontal position in feet from the start.

How can you determine where the ride reaches its highest and lowest points? Where does the track cross ground level? How many times does the height change direction? None of these questions can be answered with a linear or quadratic function. Polynomial functions, and the tools for analyzing them, make questions like this answerable.

14Sections
13Objectives
25Key Terms
12+Worked Examples

Learning Objectives

  1. 1Identify polynomial functions and describe their structure.
  2. 2Determine the degree and leading coefficient of a polynomial.
  3. 3Describe and predict the end behavior of polynomial functions.
  4. 4Find the zeros of a polynomial and interpret them as x-intercepts.
  5. 5Determine the multiplicity of a zero and describe how it affects the graph.
  6. 6Identify and count the turning points of a polynomial function.
  7. 7Sketch the graph of a polynomial function using key features.
  8. 8Divide polynomials using long division.
  9. 9Divide polynomials using synthetic division.
  10. 10Apply the Remainder Theorem and the Factor Theorem.
  11. 11Write a polynomial equation given its zeros.
  12. 12Apply polynomial functions to real-world modeling situations.
  13. 13Recognize and avoid common mistakes in polynomial function work.
Chapter 5 — Contents

Sections at a Glance

Sections 5.1 – 5.3
Structure: What Polynomials Are and How They Behave at the Edges
Definition · Degree · Leading Coefficient · End Behavior
Section 5.1

What Polynomial Functions Are

You have already worked with two families of polynomial functions without necessarily calling them that. Every linear function of the form $f(x) = mx + b$ is a polynomial. Every quadratic function of the form $f(x) = ax^2 + bx + c$ is a polynomial. Polynomial functions are simply the broader family to which both of these belong.

A polynomial function is a sum of terms, where each term consists of a coefficient multiplied by a variable raised to a whole-number power. There are no variables in denominators, no variables under radical signs, and no variables as exponents.

Formal Definition

A polynomial function in $x$ has the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $n$ is a non-negative integer (the degree), all coefficients $a_i$ are real numbers, and $a_n \neq 0$. Every exponent on the variable must be a whole number: $0, 1, 2, 3, \ldots$

Polynomial FunctionsNot Polynomial FunctionsReason Not Polynomial
$f(x) = 3x^4 - 7x^2 + 2x - 5$$f(x) = \frac{1}{x}$Variable in denominator
$g(x) = x^3$$g(x) = \sqrt{x}$Fractional exponent ($x^{1/2}$)
$h(x) = 6$$h(x) = 2^x$Variable as exponent (exponential)
$p(x) = -2x^5 + \frac{1}{2}x + 8$$p(x) = x^{-3} + x$Negative exponent

Names by Degree

DegreeNameExample
0Constant$f(x) = 7$
1Linear$f(x) = 3x - 2$
2Quadratic$f(x) = x^2 - 4x + 3$
3Cubic$f(x) = 2x^3 - x + 5$
4Quartic$f(x) = x^4 - 3x^2 + 1$
5Quintic$f(x) = x^5 + 2x^3 - x$
Domain of Polynomial Functions

Every polynomial function has a domain of all real numbers. Because there are no denominators and no radical signs with variables, any real number can be substituted for $x$. This distinguishes polynomial functions from rational functions (Chapter 7) and radical functions (Chapter 8).

Example 5.1.1
Identifying Polynomial Functions
  • ProblemDetermine whether each is a polynomial. If so, write in standard form. (a) $f(x) = 4 - 3x + x^2$   (b) $g(x) = 5x^3 - \frac{2}{x} + 1$   (c) $h(x) = -7x^4 + 3x^2 - x^{1/2}$   (d) $p(x) = 8x^6 - 3x^4 + x - 9$
  • ThinkCheck whether each term has a whole-number exponent. Rewrite $\frac{2}{x}$ as $2x^{-1}$ and $x^{1/2}$ explicitly to test the exponent.
  • Solve (a) Exponents 2, 1, 0 — all whole numbers. Polynomial. Standard form: $f(x) = x^2 - 3x + 4$

    (b) $\frac{2}{x} = 2x^{-1}$; exponent $-1$ is not a whole number. Not polynomial.

    (c) $x^{1/2}$ is a fractional exponent. Not polynomial.

    (d) Exponents 6, 4, 1, 0 — all whole numbers. Polynomial. Already in standard form.
  • CheckFor (a): can substitute any real number into $x^2 - 3x + 4$ and get a real result. No restrictions. ✓
Answer: (a) and (d) are polynomial functions. (b) and (c) are not.
Practice 5.1Concept Check
  1. What makes a function a polynomial? State the requirements in your own words.
  2. Explain why $g(x) = x^3 - \sqrt{x} + 4$ is not a polynomial function.
  3. Rewrite in standard form: (a) $3 + x^2 - 5x$    (b) $7x - x^4 + 2x^3 - 1$
  4. Give one example of a degree-4 polynomial function and one example of a function that looks polynomial but is not.
  5. Is $f(x) = 0$ a polynomial? If so, what is its degree? (Hint: look carefully at the definition.)
Section 5.2

Degree and Leading Coefficient

Once you have confirmed that a function is a polynomial, the first two features to identify are the degree and the leading coefficient. These two numbers control the most important overall behaviors of the function — particularly how the graph behaves at its far left and far right ends.

The degree is the largest exponent on the variable in standard form. The leading term is the term of highest degree. The leading coefficient is the coefficient of the leading term. The constant term is the term of degree 0, and it equals $f(0)$ — that is, the y-intercept.

Example 5.2.1
Identifying Degree, Leading Coefficient, and Constant Term
  • ProblemFor each polynomial, identify the degree, leading coefficient, and constant term.
  • ThinkRewrite in standard form if needed. Read the degree from the highest exponent, the leading coefficient from that term, and the constant from the degree-0 term.
  • Solve (a) $f(x) = 2x^4 - 5x^3 + 3x - 7$: Degree 4, leading coeff. $2$, constant $-7$

    (b) $g(x) = -x^6 + 4x^2$: Degree 6, leading coeff. $-1$, constant $0$

    (c) $h(x) = 9x - 3x^2 + 1$ → standard form $-3x^2 + 9x + 1$: Degree 2, leading coeff. $-3$, constant $1$

    (d) $p(x) = x^5$: Degree 5, leading coeff. $1$, constant $0$
  • CheckThe constant term always equals $f(0)$. Verify: $f(0) = 2(0)^4 - 5(0)^3 + 3(0) - 7 = -7$. ✓
Answer: (a) Degree 4, coeff. 2, const. −7  |  (b) Degree 6, coeff. −1, const. 0  |  (c) Degree 2, coeff. −3, const. 1  |  (d) Degree 5, coeff. 1, const. 0
Student Warning

When a polynomial is not in standard form, it is easy to misidentify the leading term. For $f(x) = 9x - 3x^2 + 1$, the first term written is $9x$, but the leading term is $-3x^2$. Always rewrite in standard form before identifying the degree and leading coefficient.

Practice 5.2Skill Practice

State the degree, leading coefficient, constant term, and y-intercept for each.

  1. $f(x) = 3x^5 - 2x^4 + x - 10$
  2. $g(x) = -4x^3 + 7$
  3. $h(x) = x^2 - 6x + 9$
  4. $p(x) = -x^7 + 5x^3 - 3x^2 + 2x$
  5. $q(x) = 12 - 4x + x^3$
  6. $r(x) = -5x^4$
Section 5.3

End Behavior

The end behavior of a polynomial describes what happens to the output values as the input grows without bound — either toward positive infinity or toward negative infinity. In graph terms, end behavior describes what the left and right ends of the graph do.

For any polynomial, when $x$ is very large, the leading term dominates all others. Higher powers grow faster. This means you only need the degree and leading coefficient to determine end behavior. There are four patterns.

DegreeLeading CoefficientAs $x \to +\infty$As $x \to -\infty$Description
EvenPositive$f(x) \to +\infty$$f(x) \to +\infty$Both ends rise
EvenNegative$f(x) \to -\infty$$f(x) \to -\infty$Both ends fall
OddPositive$f(x) \to +\infty$$f(x) \to -\infty$Left falls, right rises
OddNegative$f(x) \to -\infty$$f(x) \to +\infty$Left rises, right falls
Memory Aid

Even degree = same end behavior on both sides (both rise or both fall). Odd degree = opposite end behavior (one rises, one falls). Positive leading coefficient = right end rises. Negative leading coefficient = right end falls.

Example 5.3.1
Describing End Behavior
  • ProblemDescribe the end behavior without graphing: (a) $f(x) = -3x^5 + 2x^3 - x$   (b) $g(x) = 4x^6 - x^4 + 7$   (c) $h(x) = -2x^4 + 3x - 8$
  • ThinkIdentify the degree and leading coefficient. Apply the four patterns.
  • Solve (a) Degree 5 (odd), leading coeff. $-3$ (negative). Pattern 4: right falls, left rises.
       As $x \to +\infty$, $f(x) \to -\infty$    As $x \to -\infty$, $f(x) \to +\infty$

    (b) Degree 6 (even), leading coeff. $4$ (positive). Pattern 1: both ends rise.
       As $x \to \pm\infty$, $g(x) \to +\infty$

    (c) Degree 4 (even), leading coeff. $-2$ (negative). Pattern 2: both ends fall.
       As $x \to \pm\infty$, $h(x) \to -\infty$
  • CheckFor (a): substitute $x = 100$. The term $-3(100)^5$ is hugely negative → confirms right end falls. Substitute $x = -100$: $-3(-100)^5 = -3(-10^{10})$ → hugely positive → left end rises. ✓
Answer: (a) Falls right, rises left  |  (b) Both ends rise  |  (c) Both ends fall
Practice 5.3End Behavior Practice

For each polynomial, state the degree, leading coefficient, and describe end behavior using arrow notation.

  1. $f(x) = x^3 - 5x + 2$
  2. $g(x) = -2x^2 + 4x - 1$
  3. $h(x) = 5x^4 - 3x^3 + x$
  4. $p(x) = -x^7 + 6x^5 - x$
  5. $q(x) = x^6$
  6. $r(x) = -4x^5 + 100x^4 - 999x^3 + 1$

7. A polynomial $f(x)$ satisfies: as $x \to +\infty$, $f(x) \to +\infty$, and as $x \to -\infty$, $f(x) \to -\infty$. What can you conclude about its degree and leading coefficient?

Sections 5.4 – 5.7
Zeros, Multiplicity, Turning Points, and Graphing
Zeros · Multiplicity · Turning Points · Full Graph Strategy
Section 5.4

Zeros and x-Intercepts

A zero of a polynomial function $f(x)$ is a value of $x$ for which $f(x) = 0$. These are also called roots of the equation $f(x) = 0$, or x-intercepts of the graph. A degree-$n$ polynomial has at most $n$ real zeros.

If a polynomial $f(x)$ is written in factored form as $f(x) = (x-a)(x-b)(x-c)$, the zeros are read directly from the factors: $x = a$, $x = b$, $x = c$. To find zeros from a polynomial not yet factored, set $f(x) = 0$, factor completely, and apply the Zero Product Property.

Example 5.4.1
Finding Zeros from a Factored Polynomial
  • ProblemFind the zeros of $f(x) = (x + 3)(x - 1)(x - 5)$.
  • ThinkSet each factor equal to zero using the Zero Product Property.
  • Solve $x + 3 = 0 \Rightarrow x = -3$
    $x - 1 = 0 \Rightarrow x = 1$
    $x - 5 = 0 \Rightarrow x = 5$
  • Check$f(1) = (4)(0)(-4) = 0$ ✓
Answer: Zeros at $x = -3$, $x = 1$, $x = 5$. Graph crosses x-axis at $(-3,0)$, $(1,0)$, $(5,0)$.
Example 5.4.2
Finding Zeros by Factoring
  • ProblemFind the zeros of $f(x) = x^3 - 4x$.
  • ThinkFactor first, then set each factor equal to zero.
  • Solve $f(x) = x^3 - 4x = x(x^2 - 4) = x(x-2)(x+2)$

    $x = 0, \quad x = 2, \quad x = -2$
  • Check$f(0) = 0$ ✓   $f(2) = 8 - 8 = 0$ ✓   $f(-2) = -8 + 8 = 0$ ✓
Answer: Zeros at $x = -2$, $x = 0$, $x = 2$.
Practice 5.4Zeros Practice
  1. $f(x) = (x-4)(x+7)$
  2. $g(x) = x(x-3)(x+2)$
  3. $h(x) = (x-1)(x+1)(x-6)(x+6)$
  4. $p(x) = x^2 - 9x + 20$
  5. $q(x) = x^3 - x^2 - 6x$
  6. $r(x) = 2x^3 - 8x$
Section 5.5

Multiplicity of Zeros

The multiplicity of a zero $x = r$ is the number of times the factor $(x - r)$ appears in the fully factored form of the polynomial. Multiplicity determines how the graph behaves at that x-intercept.

Multiplicity and Graph Behavior

Odd multiplicity (1, 3, 5, …): The graph crosses the x-axis at that zero. The function changes sign.

Even multiplicity (2, 4, 6, …): The graph touches the x-axis but does not cross it. The function bounces back in the same direction.

The sum of all multiplicities equals the degree of the polynomial.

Example 5.5.1
Identifying Multiplicity and Graph Behavior
  • ProblemFor $f(x) = (x+3)^2(x-1)(x-4)^3$, identify all zeros, multiplicities, and graph behavior at each.
  • ThinkRead the multiplicity from the exponent on each factor. Apply the odd/even rule.
  • Solve $x = -3$: multiplicity 2 (even) → touches, does not cross
    $x = 1$: multiplicity 1 (odd) → crosses
    $x = 4$: multiplicity 3 (odd) → crosses (with a flattening)
  • CheckTotal degree: $2 + 1 + 3 = 6$. This is a degree-6 polynomial with 3 distinct zeros. Multiplicities sum correctly. ✓
Answer: Touches at $x = -3$; crosses at $x = 1$; crosses (flattened S-shape) at $x = 4$.
Diagram Instruction

Draw three small diagrams showing what the graph looks like near an x-intercept for: (a) Multiplicity 1 — a simple straight crossing through the x-axis; (b) Multiplicity 2 — a parabola-like touch that bounces back; (c) Multiplicity 3 — an S-shaped crossing that flattens as it passes through.

Practice 5.5Multiplicity Practice

For each polynomial, identify all zeros, their multiplicities, and graph behavior at each zero.

  1. $f(x) = (x-5)^2(x+2)$
  2. $g(x) = x^3(x-3)^2(x+1)$
  3. $h(x) = (x+4)^2(x-1)^2$
  4. $p(x) = -(x-2)^3(x+5)$
  5. $q(x) = x^2(x^2 - 9)$
  6. Write a degree-5 polynomial with zero at $x=0$ (mult. 3) and $x=-2$ (mult. 2), leading coefficient 1.
Section 5.6

Turning Points

A turning point is a point where the graph changes from increasing to decreasing, or vice versa. A local maximum is a peak; a local minimum is a valley. A polynomial of degree $n$ has at most $n - 1$ turning points.

DegreeMax Turning Points
1 (linear)0
2 (quadratic)1
3 (cubic)2
4 (quartic)3
$n$$n - 1$
Finding Turning Points

Finding exact coordinates of turning points requires calculus. In Algebra II, use tables of values and graphing technology to estimate them. Use the $n - 1$ maximum as a guide to the complexity of the graph.

Practice 5.6Turning Points
  1. What is the maximum number of turning points for a degree-7 polynomial?
  2. A polynomial has 4 turning points. What is the minimum possible degree?
  3. For $f(x) = x^3 - 3x$, evaluate $f$ at $x = -2, -1, 0, 1, 2$ and estimate the locations of any turning points.
  4. A polynomial graph rises from the left, reaches a local max, falls to a local min, then rises again. What is the minimum degree?
Section 5.7

Graphing Polynomial Functions

A complete sketch of a polynomial function requires five elements: end behavior, the y-intercept, zeros, multiplicity behavior at each zero, and a sign chart or table to determine whether the graph is above or below the x-axis between zeros.

Graphing Strategy — Five Steps

Step 1: Determine end behavior (degree and leading coefficient).

Step 2: Find the y-intercept ($f(0)$).

Step 3: Find zeros and their multiplicities.

Step 4: Build a sign chart: test a value in each interval between zeros.

Step 5: Sketch — start with end behavior, mark intercepts, draw curve through all information.

Example 5.7.1
Graphing a Polynomial Function
  • ProblemSketch the graph of $f(x) = -x(x-2)^2(x+3)$.
  • ThinkWork through all five steps systematically.
  • Solve End behavior: Leading term is $-x \cdot x^2 \cdot x = -x^4$. Degree 4, negative → both ends fall.

    y-intercept: $f(0) = -(0)(4)(3) = 0$. The y-intercept is at $(0, 0)$.

    Zeros: $x = 0$ (mult. 1, crosses), $x = 2$ (mult. 2, touches), $x = -3$ (mult. 1, crosses).

    Sign chart:
    Test $x = -4$: $f(-4) = -(-4)(36)(-1) = -144$ → negative (below x-axis)
    Test $x = -1$: $f(-1) = (1)(9)(2) = 18$ → positive (above x-axis)
    Test $x = 1$: $f(1) = -(1)(1)(4) = -4$ → negative (below x-axis)
    Test $x = 3$: $f(3) = -(3)(1)(6) = -18$ → negative (below x-axis)
  • CheckDegree 4, total multiplicity $1+2+1 = 4$ ✓. Both ends fall ✓. Sign pattern consistent with crossings and touching behavior ✓.
Answer: The graph falls from lower-left, crosses up at $x = -3$, rises through $(-3, 0)$, crosses down at $x = 0$ (the y-intercept), falls below zero, touches the x-axis at $x = 2$ and bounces back down, then falls to the lower-right.
Diagram Instruction

Draw the coordinate plane with x-intercepts marked at $-3$, $0$, and $2$. The curve rises from lower-left, crosses upward through $(-3,0)$, reaches a peak above the x-axis, crosses downward through $(0,0)$, falls below zero, touches the x-axis at $(2,0)$ with a parabolic bounce, and falls to the lower-right. Both ends point downward.

Practice 5.7Graphing Practice

For each polynomial, determine end behavior, zeros and multiplicities, y-intercept, sign behavior, and sketch the graph.

  1. $f(x) = x(x-3)(x+2)$
  2. $g(x) = -(x+1)^2(x-4)$
  3. $h(x) = x^2(x-2)(x+1)$
  4. $p(x) = (x-1)^3(x+2)$
  5. $q(x) = -2(x+3)(x-1)(x-5)$
Sections 5.8 – 5.9
Polynomial Division: Long Division and Synthetic Division
Long Division · Synthetic Division · Division Algorithm
Section 5.8

Polynomial Long Division

When a polynomial is not easy to factor, dividing by a known factor reveals the remaining structure. The Division Algorithm for Polynomials states that for any polynomial $f(x)$ and divisor $d(x)$:

Division Algorithm

$f(x) = d(x) \cdot q(x) + r(x)$

where $q(x)$ is the quotient, $r(x)$ is the remainder, and the degree of $r(x)$ is less than the degree of $d(x)$.

Placeholder Rule

Always include a zero-coefficient term for any missing degree. Dividing $x^3 - 8$ requires writing it as $x^3 + 0x^2 + 0x - 8$. Skipping placeholders shifts columns and produces wrong answers.

Example 5.8.1
Polynomial Long Division
  • ProblemDivide $f(x) = 2x^3 - 3x^2 + x - 5$ by $d(x) = x - 2$.
  • ThinkDivide leading term of dividend by leading term of divisor at each step. Subtract, bring down, repeat.
  • Solve Step 1: $2x^3 \div x = 2x^2$. Multiply: $2x^2(x-2) = 2x^3 - 4x^2$. Subtract → remainder $x^2 + x - 5$.

    Step 2: $x^2 \div x = x$. Multiply: $x(x-2) = x^2 - 2x$. Subtract → remainder $3x - 5$.

    Step 3: $3x \div x = 3$. Multiply: $3(x-2) = 3x - 6$. Subtract → remainder $1$.

    Degree of remainder (0) < degree of divisor (1). Stop.
  • Check$(x-2)(2x^2 + x + 3) + 1 = 2x^3 + x^2 + 3x - 4x^2 - 2x - 6 + 1 = 2x^3 - 3x^2 + x - 5$ ✓
$$\frac{2x^3 - 3x^2 + x - 5}{x - 2} = 2x^2 + x + 3 + \frac{1}{x-2}$$
Example 5.8.2
Long Division with a Missing Term
  • ProblemDivide $x^3 - 8$ by $x - 2$.
  • ThinkInsert placeholders: $x^3 + 0x^2 + 0x - 8$.
  • Solve Step 1: $x^2$  ·  $(x-2) = x^3 - 2x^2$  →  new: $2x^2 + 0x - 8$
    Step 2: $2x$  ·  $(x-2) = 2x^2 - 4x$  →  new: $4x - 8$
    Step 3: $4$  ·  $(x-2) = 4x - 8$  →  remainder $0$
  • Check$(x-2)(x^2 + 2x + 4) = x^3 - 8$ ✓
Answer: $x^3 - 8 = (x-2)(x^2 + 2x + 4)$. Remainder 0 confirms $x = 2$ is a zero.
Practice 5.8Long Division Practice
  1. $(x^2 + 7x + 10) \div (x+2)$
  2. $(x^3 - 2x^2 - 5x + 6) \div (x-3)$
  3. $(2x^3 + x^2 - 7x + 3) \div (x+2)$
  4. $(x^4 - 16) \div (x-2)$
  5. $(3x^3 - 5x + 2) \div (x^2 - 1)$
Section 5.9

Synthetic Division

When the divisor has the form $(x - c)$ — a linear factor with leading coefficient 1 — synthetic division provides a streamlined method using only the coefficients of the dividend and the value $c$. It produces identical results to long division, but faster.

Synthetic Division Procedure

1. Write $c$ in a box to the left. Write all coefficients of $f(x)$ in order (include zeros for missing degrees).

2. Bring down the first coefficient.

3. Multiply by $c$; write under the next coefficient; add. Repeat to the end.

4. The last number is the remainder. The other numbers are quotient coefficients.

Sign of c

When dividing by $(x + 4)$, rewrite as $(x - (-4))$, so use $c = -4$, not $c = 4$. Using the wrong sign gives an entirely wrong result.

Example 5.9.1
Synthetic Division
  • ProblemDivide $f(x) = 3x^3 - 5x^2 + 2x + 7$ by $(x - 2)$. Use $c = 2$.
  • ThinkCoefficients are $3, -5, 2, 7$. Set up the array with $c = 2$.
  • Solve 
      2 | 3   -5    2    7
    |      6    2    8
    ——————————————————
      3    1    4   15
    Remainder: $15$. Quotient: $3x^2 + x + 4$.
  • Check$(x-2)(3x^2+x+4)+15 = 3x^3-5x^2+2x+7$ ✓
$$\frac{3x^3 - 5x^2 + 2x + 7}{x - 2} = 3x^2 + x + 4 + \frac{15}{x-2}$$
Example 5.9.2
Synthetic Division with Zero Remainder
  • ProblemDivide $f(x) = x^3 + 4x^2 - x - 4$ by $(x + 1)$. Use $c = -1$.
  • Solve 
     -1 | 1    4   -1   -4
    |     -1   -3    4
    ——————————————————
      1    3   -4    0
    Remainder: $0$. Quotient: $x^2 + 3x - 4$.
  • Check$(x+1)(x^2+3x-4) = x^3+4x^2-x-4$ ✓
Answer: $x^3 + 4x^2 - x - 4 = (x+1)(x^2+3x-4)$. Remainder 0 means $x = -1$ is a zero.
Practice 5.9Synthetic Division Practice
  1. $(x^2 + 5x + 6) \div (x+2)$
  2. $(x^3 - 7x + 6) \div (x-2)$
  3. $(2x^3 + 3x^2 - 11x - 6) \div (x+3)$
  4. $(x^4 - 5x^2 + 4) \div (x-1)$
  5. $(x^3 + 27) \div (x+3)$
Sections 5.10 – 5.12
The Remainder Theorem, Factor Theorem, and Building Polynomials from Zeros
Remainder Theorem · Factor Theorem · Polynomial Equations from Zeros
Section 5.10

The Remainder Theorem

Remainder Theorem

If a polynomial $f(x)$ is divided by $(x - c)$, then the remainder equals $f(c)$.

Why: The Division Algorithm gives $f(x) = (x-c) \cdot q(x) + r$. Substituting $x = c$: $f(c) = (c-c) \cdot q(c) + r = 0 + r = r$.

This theorem turns function evaluation into a division problem. Instead of substituting $c$ into a complicated polynomial expression, perform synthetic division and read $f(c)$ as the remainder. This is faster for high-degree polynomials and immediately tells you whether $c$ is a zero.

Example 5.10.1
Using the Remainder Theorem
  • ProblemFind $f(3)$ for $f(x) = x^4 - 2x^3 + 5x - 7$, using the Remainder Theorem.
  • ThinkDivide $f(x)$ by $(x-3)$ using synthetic division. Include 0 for the missing $x^2$ term. The remainder is $f(3)$.
  • Solve Coefficients: $1, -2, 0, 5, -7$ 
      3 | 1   -2    0    5   -7
    |      3    3    9   42
    —————————————————————————
      1    1    3   14   35
    Remainder: $35$
  • CheckDirect substitution: $f(3) = 81 - 54 + 15 - 7 = 35$ ✓
Answer: $f(3) = 35$
Practice 5.10Remainder Theorem

Use the Remainder Theorem (synthetic division) to find each function value.

  1. $f(x) = x^3 - 4x + 1$; find $f(2)$
  2. $g(x) = 2x^4 - x^3 + 3x - 5$; find $g(-1)$
  3. $h(x) = x^5 - 32$; find $h(2)$
  4. $p(x) = 3x^3 + 2x^2 - x + 4$; find $p(-3)$
  5. A polynomial divided by $(x-5)$ gives remainder $-7$. What is $f(5)$?
Section 5.11

The Factor Theorem

Factor Theorem

$(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$.

Knowing any one of the following tells you all three: that $c$ is a zero of $f$, that $f(c) = 0$, or that $(x - c)$ is a factor of $f(x)$.

Example 5.11.1
Applying the Factor Theorem
  • ProblemDetermine whether $(x-3)$ is a factor of $f(x) = x^3 - 5x^2 + 3x + 9$. If so, find the complete factored form.
  • Think$(x-3)$ is a factor if and only if $f(3) = 0$. Use synthetic division with $c = 3$.
  • Solve 
      3 | 1   -5    3    9
    |      3   -6   -9
    ——————————————————
      1   -2   -3    0
    Remainder = 0 → $f(3) = 0$ → $(x-3)$ is a factor.
    Quotient: $x^2 - 2x - 3 = (x-3)(x+1)$
    Complete factored form: $f(x) = (x-3)^2(x+1)$
  • Check$f(3) = 27 - 45 + 9 + 9 = 0$ ✓   $f(-1) = -1 - 5 - 3 + 9 = 0$ ✓
Answer: Yes, $(x-3)$ is a factor. $f(x) = (x-3)^2(x+1)$.
Example 5.11.2
Using a Known Zero to Build Factored Form
  • ProblemGiven that $x = -2$ is a zero of $g(x) = 2x^3 + x^2 - 8x - 4$, find the complete factored form.
  • ThinkSince $x = -2$ is a zero, $(x+2)$ is a factor. Divide by $(x+2)$ using synthetic division with $c = -2$.
  • Solve 
     -2 | 2    1   -8   -4
    |     -4    6    4
    ——————————————————
      2   -3   -2    0
    Quotient: $2x^2 - 3x - 2 = (2x+1)(x-2)$
    Complete factored form: $g(x) = (x+2)(2x+1)(x-2)$
  • Check$g(-2) = 2(-8) + 4 + 16 - 4 = 0$ ✓
Answer: $g(x) = (x+2)(2x+1)(x-2)$, with zeros at $x = -2$, $x = -\frac{1}{2}$, $x = 2$.
Practice 5.11Factor Theorem
  1. Use the Factor Theorem to determine whether $(x-4)$ is a factor of $f(x) = x^3 - 3x^2 - 10x + 24$.
  2. Use the Factor Theorem to determine whether $(x+2)$ is a factor of $g(x) = x^3 + x^2 - 4x - 4$.
  3. Given $x = 1$ is a zero of $h(x) = x^3 - 2x^2 - 5x + 6$, find the complete factored form.
  4. Given $x = -3$ is a zero of $p(x) = x^3 + 4x^2 + x - 6$, find all zeros and write the factored form.
  5. Explain in your own words why the Factor Theorem follows directly from the Remainder Theorem.
Section 5.12

Finding Polynomial Equations from Zeros

The factor-zero connection works in reverse: given a set of zeros (with their multiplicities), you can build the polynomial. If $r_1, r_2, \ldots, r_n$ are the zeros with multiplicities $m_1, m_2, \ldots, m_n$, then:

$$f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2} \cdots (x - r_n)^{m_n}$$

The leading coefficient $a$ can be any nonzero real number unless additional information (such as a specific point on the graph) is given to determine it exactly.

Example 5.12.1
Writing a Polynomial from Zeros
  • ProblemWrite a polynomial of least degree with zeros at $x = -1$, $x = 3$, and $x = 5$, with leading coefficient 1.
  • ThinkEach zero $r$ contributes factor $(x - r)$. Multiply them.
  • Solve $f(x) = (x+1)(x-3)(x-5)$
    $(x+1)(x-3) = x^2 - 2x - 3$
    $(x^2 - 2x - 3)(x-5) = x^3 - 7x^2 + 7x + 15$
Answer: $f(x) = x^3 - 7x^2 + 7x + 15$
Example 5.12.2
Using a Known Point to Find the Leading Coefficient
  • ProblemWrite a polynomial with zeros at $x = 0$ (mult. 2) and $x = -4$ (mult. 1), given that $f(2) = 24$.
  • ThinkBuild the general form with unknown $a$, then use the given point.
  • Solve $f(x) = a \cdot x^2(x+4)$
    $f(2) = a(4)(6) = 24a = 24 \Rightarrow a = 1$
    $f(x) = x^2(x+4) = x^3 + 4x^2$
  • Check$f(2) = 4(6) = 24$ ✓. Zeros: $x = 0$ (mult. 2, touches), $x = -4$ (mult. 1, crosses). ✓
Answer: $f(x) = x^3 + 4x^2$
Practice 5.12Writing Polynomial Equations
  1. Write a degree-3 polynomial with zeros at $x = 2$, $x = -3$, $x = 7$, leading coefficient 1.
  2. Write a polynomial with zeros at $x = 0$, $x = 1$, $x = -1$ (each mult. 1), leading coefficient $-2$.
  3. Write a polynomial with zeros at $x = 4$ (mult. 2) and $x = -1$ (mult. 1), leading coefficient 1.
  4. A degree-4 polynomial has zeros at $x = 3$ and $x = -2$, each with multiplicity 2. Write it with leading coefficient 1.
  5. A degree-3 polynomial has zeros at $x = 1$, $x = -1$, $x = 5$ and passes through $(0, -10)$. Find the polynomial.
Sections 5.13 – 5.14
Models in Context and Common Mistakes
Polynomial Models · Real-World Applications · Ten Common Errors
Section 5.13

Polynomial Models in Context

Polynomial functions appear throughout applied mathematics because they can rise, fall, change direction multiple times, and model curved data that linear or quadratic functions cannot capture. Applications include physics (projectile and beam models), engineering (stress curves, deflection), economics (revenue and profit), biology (population curves), and computer graphics (splines).

Example 5.13.1
Profit as a Polynomial Function
  • ProblemA company's weekly profit (in hundreds of dollars) is $P(x) = -2x^3 + 18x^2 - 40x$, where $x$ is thousands of units produced, $0 \leq x \leq 6$. (a) Find $P(0)$ and interpret. (b) Find zeros and interpret. (c) For what values of $x$ is the company profitable?
  • ThinkRead → Represent → Solve → Check → Explain. Factor to find zeros; test the sign of $P(x)$ between them.
  • Solve (a) $P(0) = 0$. Producing nothing yields no profit.

    (b) $P(x) = -2x(x^2 - 9x + 20) = -2x(x-4)(x-5)$
    Zeros at $x = 0$, $x = 4$, $x = 5$.
    $x = 4$: producing 4,000 units yields zero profit (breakeven).
    $x = 5$: producing 5,000 units also breaks even.

    (c) Test $x = 4.5$: $P(4.5) = -2(4.5)(0.5)(-0.5) = 2.25 > 0$
    Test $x = 2$: $P(2) = -2(2)(-2)(-3) = -24 < 0$
    The company is profitable when $4 < x < 5$.
  • ExplainToo few units → losses (revenue doesn't cover costs). A narrow range (4,000–5,000 units) is profitable. Too many units → losses again from inefficiency or excess costs.
Answer: The company is profitable when production is between 4,000 and 5,000 units per week.
Example 5.13.2
Volume as a Polynomial Model
  • ProblemA 20-inch by 16-inch sheet of cardboard has equal squares of side $x$ cut from each corner, then folded into an open box. Write volume as a polynomial function of $x$.
  • ThinkAfter cutting: length $= 20 - 2x$, width $= 16 - 2x$, height $= x$. Domain: $0 < x < 8$ (width must be positive).
  • Solve $V(x) = x(20-2x)(16-2x)$
    $= x(4x^2 - 72x + 320)$
    $= 4x^3 - 72x^2 + 320x$
  • Check$V(0) = 0$ (no height) ✓   $V(8) = 4(512) - 72(64) + 320(8) = 2048 - 4608 + 2560 = 0$ (no material) ✓
Answer: $V(x) = 4x^3 - 72x^2 + 320x$ for $0 < x < 8$.
Practice 5.13Modeling Problems
  1. A projectile's height in feet is $h(t) = -16t^2 + 48t + 64$, where $t$ is time in seconds. (a) Find $h(0)$ and interpret it. (b) Find the positive zero and interpret it. (c) Build a table of values to estimate the maximum height.
  2. A rectangular garden has a perimeter of 60 feet. Area as a function of width $x$ is $A(x) = x(30-x)$. (a) Write in standard form. (b) Find zeros and interpret them. (c) What width maximizes area?
  3. A beam's deflection in mm is $D(x) = 0.001x(x-10)(x-20)$, where $x$ is position in meters, $0 \leq x \leq 20$. (a) Find all zeros and interpret. (b) Find $D(5)$ and interpret.
Section 5.14

Common Polynomial Function Mistakes

Mistake 1 — Misidentifying Degree from Non-Standard Form
For $f(x) = 3x^2 + 5x^4 - x + 7$, saying degree is 2 (first term written)

Correct: Rewrite in standard form first — $5x^4 + 3x^2 - x + 7$. The degree is 4.

Mistake 2 — Getting the Leading Coefficient Wrong
For $f(x) = 4 - 3x^5$, saying leading coefficient is 4

Correct: Standard form is $-3x^5 + 4$. Leading coefficient is $-3$.

Mistake 3 — Reversing Even/Odd End Behavior
Saying a degree-5 polynomial with positive leading coefficient has both ends rising

Correct: Odd degree means opposite ends. Positive leading coefficient → right end rises, left end falls.

Mistake 4 — Forgetting Placeholders in Division
Dividing $x^3 - 8$ by $(x-2)$ with only two coefficients: $1, -8$

Correct: Include all coefficients including zeros: $1, 0, 0, -8$.

Mistake 5 — Wrong Sign for $c$ in Synthetic Division
Dividing by $(x+4)$ and using $c = 4$

Correct: $(x+4) = (x - (-4))$, so $c = -4$.

Mistake 6 — Confusing Multiplicity and Graph Behavior
"The graph crosses at a zero with even multiplicity."

Correct: Even multiplicity → touches and bounces. Odd multiplicity → crosses.

Mistake 7 — Claiming Exactly $n$ Real Zeros
"A degree-4 polynomial must have exactly 4 zeros."

Correct: A degree-4 polynomial has at most 4 real zeros — it may have 0, 1, 2, 3, or 4, with remaining zeros being complex.

Mistake 8 — Skipping the GCF Before Factoring
Trying to factor $2x^3 - 8x$ by trinomial methods directly

Correct: Factor out GCF first: $2x^3 - 8x = 2x(x^2 - 4) = 2x(x-2)(x+2)$.

Mistake 9 — Applying the Factor Theorem When Remainder Is Nonzero
After getting remainder 7 from synthetic division, concluding $c$ is a zero

Correct: $c$ is a zero only if the remainder is exactly 0. A nonzero remainder means $c$ is not a zero.

Mistake 10 — Using the Wrong Point for the y-Intercept
Evaluating $f(1)$ and calling it the y-intercept

Correct: The y-intercept is always $f(0)$, which equals the constant term of the polynomial.

Chapter 5 — Back Matter

Chapter Summary

Key Terms Review

Polynomial function
A function of the form $a_n x^n + \cdots + a_0$ where all exponents are non-negative integers and $a_n \neq 0$.
Degree
The largest exponent on the variable. Controls the maximum number of zeros and turning points.
Leading term
The term with the highest degree. Determines end behavior for large $|x|$.
Leading coefficient
The coefficient of the leading term. Controls whether the right end of the graph rises or falls.
Constant term
The degree-0 term; equals $f(0)$ and gives the y-intercept.
Standard form
A polynomial written with terms in order from highest to lowest degree.
End behavior
What $f(x)$ does as $x \to +\infty$ or $x \to -\infty$. Determined by degree and leading coefficient.
Zero
A value $c$ where $f(c) = 0$. Also called a root, solution, or x-intercept.
Multiplicity
How many times a factor $(x - c)$ appears. Determines whether the graph crosses or touches the x-axis.
Turning point
A point where the graph changes direction. A degree-$n$ polynomial has at most $n - 1$.
Local maximum
A turning point that is a peak — higher than nearby points, but not necessarily the global highest.
Local minimum
A turning point that is a valley — lower than nearby points.
Long division
A method for dividing any polynomial by any polynomial divisor, using a process analogous to numerical long division.
Synthetic division
A shorthand method for dividing by a linear factor $(x - c)$, operating on coefficients alone.
Remainder Theorem
When $f(x)$ is divided by $(x - c)$, the remainder equals $f(c)$.
Factor Theorem
$(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. Connects factors, zeros, and roots.
Dividend
The polynomial being divided.
Divisor
The polynomial you are dividing by.
Quotient
The result of division, not including the remainder.
Remainder
What is left after division; its degree is less than the divisor's degree.

Concept Review Questions

  1. In your own words, what makes a function a polynomial? What would disqualify it?
  2. Why does the degree determine the maximum number of x-intercepts?
  3. Explain the four end behavior patterns. What two features determine them?
  4. What is the difference between odd and even multiplicity at a zero? How does each affect the graph?
  5. A degree-4 polynomial has "at most" 3 turning points. Give an example with fewer than 3.
  6. Explain the Remainder Theorem in your own words. Why is it true?
  7. State the Factor Theorem. How does it follow from the Remainder Theorem?
  8. If you know the zeros of a polynomial, what additional information do you need to determine the polynomial completely?
  9. Give one real-world situation where a polynomial model is better than linear or quadratic, and explain why.
  10. Why must you include zero-coefficient placeholders in polynomial division?

Skill Practice

Part AEnd Behavior
  1. $f(x) = 7x^3 - 2x$
  2. $g(x) = -4x^4 + x^2 + 8$
  3. $h(x) = x^5 - 100x^4$
  4. $p(x) = -x^6 + 3x^5 - x + 9$
  5. $q(x) = 2x^8$
Part BZeros and Multiplicity

Find all zeros and their multiplicities. State whether graph crosses or touches at each.

  1. $f(x) = (x-2)^3(x+5)$
  2. $g(x) = x^2(x-3)^2(x+1)$
  3. $h(x) = 3x(x+4)^2$
  4. $p(x) = -(x-1)^4(x-7)$
  5. $q(x) = x^4 - 16$
Part CGraphing

Sketch each function. Show end behavior, zeros, multiplicity behavior, and y-intercept.

  1. $f(x) = x(x-4)(x+2)$
  2. $g(x) = -(x+1)^2(x-3)$
  3. $h(x) = x^2(x-2)(x+2)$
  4. $p(x) = (x-1)^3$
Part DPolynomial Division
  1. $(x^2 + 3x - 10) \div (x-2)$
  2. $(x^3 + 2x^2 - 11x - 12) \div (x+4)$
  3. $(2x^3 - 7x + 3) \div (x^2 + x - 1)$
  4. $(x^4 - 1) \div (x-1)$
  5. $(3x^3 + 5x^2 - x - 2) \div (x+2)$
Part ERemainder and Factor Theorem
  1. Find $f(-2)$ for $f(x) = x^4 - 3x^2 + x - 5$.
  2. Is $(x-3)$ a factor of $g(x) = x^3 - 7x + 6$?
  3. Is $(x+1)$ a factor of $h(x) = 2x^4 + x^3 - x - 2$?
  4. Given $x = 2$ is a zero of $p(x) = x^3 - 6x^2 + 11x - 6$, find all zeros.
  5. Given $x = -3$ is a zero of $q(x) = x^3 + 5x^2 + 3x - 9$, find the factored form.
Part FWriting Polynomials
  1. Zeros at $x=1$, $x=-2$, $x=4$; lead. coeff. 1
  2. Zeros at $x=0$ (mult. 3) and $x=5$ (mult. 1); lead. coeff. $-1$
  3. Zeros at $x=-1$ and $x=3$ (each mult. 2); $f(0)=18$

Word Problems

  1. A ball is thrown upward from a 100-foot building. Its height is $h(t) = -16t^2 + 32t + 100$ feet after $t$ seconds. When does the ball hit the ground?
  2. A box is formed from a 30 cm by 24 cm sheet of cardboard by cutting $x$-cm squares from each corner and folding. Write the volume function. State the domain.
  3. A company's monthly profit (in thousands) is $P(x) = -x^3 + 9x^2 - 15x - 3$, where $x$ is the number of product lines (in dozens), $0 \leq x \leq 8$. Find $P(3)$, $P(5)$, and $P(7)$. Describe how profit changes.
  4. A cable between two poles follows $h(x) = 0.01x^2 - 0.6x + 12$, where $x$ is horizontal position in feet and $h$ is height. Find the lowest point of the cable (the vertex of the parabola).
  5. The population of bacteria (thousands) over time $t$ hours follows $N(t) = t^4 - 10t^2 + 9$, for $0 \leq t \leq 3$. Find all zeros in the domain and interpret. Find $N(1)$ and $N(2)$.

Error Analysis

Each problem contains a mistake. Identify the error and show the correct work.

Problem 1
For $f(x) = -x^5 + 3x^2 - 7$: "degree is 2, leading coefficient is $-1$."

What is wrong? State the correct degree and leading coefficient.

Problem 2
For $f(x) = 3x^4 - 2x^3 + 1$: "right end falls, left end rises."

What is the error? State the correct end behavior.

Problem 3
Dividing $x^3 - 4x + 3$ by $(x-2)$ with synthetic array $1, -4, 3$ (only 3 coefficients).

Identify the error. Show the correct setup and result.

Problem 4
For $f(x) = (x-3)^2(x+1)$: "the graph crosses at $x = 3$ because the exponent is 2."

What is wrong? Describe the correct graph behavior at $x = 3$.

Problem 5
Writing a polynomial with zeros at $x = 2$, $x = -5$, $x = 4$ as $f(x) = (x+2)(x-5)(x+4)$.

Identify the error in every factor. Write the correct polynomial.

Chapter 5 Checkpoint

Chapter 5 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 5
Section A — Polynomial Identification and Features
  1. Determine whether $f(x) = 3x^2 - \sqrt{x} + 1$ is a polynomial. Explain.
  2. For $g(x) = -4x^5 + 3x^3 - x + 9$, identify: (a) degree, (b) leading coefficient, (c) constant term, (d) y-intercept.
  3. Describe the end behavior of $h(x) = 2x^6 - 7x^4 + 1$.
  4. Describe the end behavior of $p(x) = -3x^7 + 100x^6$.
Section B — Zeros, Multiplicity, and Graphs
  1. Find all zeros of $f(x) = x(x+3)^2(x-5)$ and state their multiplicities and graph behavior.
  2. Sketch the graph of $g(x) = -(x-2)(x+1)^2$. Show all key features.
  3. A degree-6 polynomial has zeros at $x = 0$ (mult. 3), $x = 4$ (mult. 2), and $x = -1$ (mult. 1). What is the total degree? What is the maximum number of turning points?
Section C — Polynomial Division
  1. Divide $(x^3 - 5x^2 + 3x + 9)$ by $(x - 3)$ using synthetic division.
  2. Divide $(2x^3 + x^2 - 5x + 2)$ by $(x + 2)$ using synthetic division.
  3. Divide $(x^4 - 3x^2 + 5)$ by $(x^2 - 1)$ using long division.
Section D — Remainder and Factor Theorems
  1. Use the Remainder Theorem to find $f(-2)$ for $f(x) = x^3 - x^2 + 4x - 8$.
  2. Determine whether $(x - 1)$ is a factor of $g(x) = 3x^4 - 5x^3 + 2x^2 - x + 1$.
  3. Given that $x = 3$ is a zero of $h(x) = x^3 - 4x^2 - x + 4$, find all zeros and write the complete factored form.
Section E — Writing and Applying Polynomials
  1. Write a polynomial of least degree with zeros at $x = -4$, $x = 0$, and $x = 6$, leading coefficient 1.
  2. Write a polynomial with zeros at $x = 2$ (mult. 2) and $x = -3$ (mult. 1), and with $f(0) = -36$.
  3. A box is formed from a 20-inch by 12-inch cardboard sheet by cutting $x$-inch squares from each corner. Write the volume function and state the domain.

Answers may be placed in the answer key.

Looking Ahead
Chapter 6: Polynomial Equations and Theorems

Chapter 5 gave you the tools to understand polynomial functions — structure, graphs, zeros, and the Remainder and Factor Theorems. Chapter 6 takes the next step: solving polynomial equations systematically, especially when factoring is not immediately obvious.

  • The Rational Root Theorem — generating a list of possible rational zeros to test
  • Testing rational root candidates with synthetic division and the Factor Theorem
  • Depressed polynomials — reducing degree step by step until complete factorization
  • Complex roots and conjugate pairs — what happens when polynomial equations have non-real solutions
  • The Fundamental Theorem of Algebra — every degree-$n$ polynomial has exactly $n$ roots (counting multiplicity and complex roots)
  • Counting roots with multiplicity and using graphing technology to approximate irrational roots

Every skill from Chapter 5 — factoring, synthetic division, the Remainder Theorem, the Factor Theorem, and writing polynomials from zeros — is exactly the foundation Chapter 6 requires.

MGU Library Connection
  • Appendix F: Polynomial Factoring and Division Reference — condensed summary of all factoring patterns, long division steps, and synthetic division templates
  • Appendix B: Function Parent Graphs — visual references of polynomial graph shapes by degree
  • Chapter 6: Polynomial Equations and Theorems — Rational Root Theorem, Fundamental Theorem of Algebra, complex roots
  • Calculus I — exact turning points are found using derivatives; the Remainder Theorem reappears in the context of Taylor polynomials and polynomial approximation
  • MGU Dictionary entries: polynomial, zero, multiplicity, end behavior, synthetic division, Remainder Theorem, Factor Theorem