Polynomial Functions

If you have ever watched a roller coaster climb, drop, twist, and turn, you have seen something that behaves remarkably like a polynomial function. These mathematical expressions can rise and fall multiple times, creating curves that model everything from the arc of a basketball to the profit margins of a business over time. While linear functions give you straight lines and quadratic functions give you simple parabolas, polynomial functions open up a whole world of more complex and interesting shapes.

The good news is that you already understand the building blocks. If you can work with expressions like $x^2$ or $x^3$, then you are ready to tackle polynomials. In this chapter, we will explore what happens when you combine these powers together, how to predict the behavior of these functions, and how to find where they cross the x-axis. By the end, you will be able to look at a polynomial and understand its personality before you even graph it.

Core Concepts

What Is a Polynomial Function?

A polynomial function is a function that consists of terms where a variable is raised to non-negative integer powers and multiplied by coefficients. The general form looks like this:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$$

That notation might look intimidating, but let us break it down with a concrete example:

$$f(x) = 2x^4 - 5x^3 + x^2 - 7x + 3$$

This is a polynomial function. Each piece (like $2x^4$ or $-7x$) is called a term. The numbers in front of the variables (2, -5, 1, -7, and 3) are called coefficients.

Here is what makes something a polynomial:

• All exponents on the variable must be whole numbers (0, 1, 2, 3, …)
• No variables in denominators (so $\frac{1}{x}$ is not allowed)
• No variables under radicals (so $\sqrt{x}$ is not allowed)

Degree and Leading Coefficient

Two features tell you the most about a polynomial’s behavior:

The degree is the highest exponent that appears on the variable. In $f(x) = 2x^4 - 5x^3 + x^2 - 7x + 3$, the degree is 4 because $x^4$ is the highest power.

The leading coefficient is the coefficient of the term with the highest degree. In our example, the leading coefficient is 2 (from the $2x^4$ term).

The leading term is the term with the highest degree. Here, it is $2x^4$.

These two numbers, the degree and leading coefficient, are like a polynomial’s fingerprint. They tell you how the function behaves when $x$ gets very large or very small.

Standard Form

A polynomial is in standard form when its terms are written in order from highest degree to lowest degree:

$$f(x) = 2x^4 - 5x^3 + x^2 - 7x + 3$$

This is standard form. The powers decrease as you read from left to right: 4, 3, 2, 1, 0.

Sometimes you will encounter a polynomial written out of order, like:

$$g(x) = 3 - x + 4x^3 + x^2$$

Rewriting this in standard form gives you:

$$g(x) = 4x^3 + x^2 - x + 3$$

Standard form makes it easy to identify the degree (3) and leading coefficient (4) at a glance.

End Behavior

End behavior describes what happens to the function values as $x$ approaches positive infinity ($x \to +\infty$) and negative infinity ($x \to -\infty$). Think of it as asking: “What does the graph do way out on the far left and far right?”

The end behavior depends only on the leading term because when $x$ is very large (or very negative), the highest power dominates all the others.

Here are the four patterns based on degree and leading coefficient:

Even degree, positive leading coefficient:

• As $x \to +\infty$, $f(x) \to +\infty$ (rises on the right)
• As $x \to -\infty$, $f(x) \to +\infty$ (rises on the left)
• Shape: like a “U” or a bowl opening upward

Even degree, negative leading coefficient:

• As $x \to +\infty$, $f(x) \to -\infty$ (falls on the right)
• As $x \to -\infty$, $f(x) \to -\infty$ (falls on the left)
• Shape: like an upside-down “U” or a bowl opening downward

Odd degree, positive leading coefficient:

• As $x \to +\infty$, $f(x) \to +\infty$ (rises on the right)
• As $x \to -\infty$, $f(x) \to -\infty$ (falls on the left)
• Shape: rising from lower left to upper right

Odd degree, negative leading coefficient:

• As $x \to +\infty$, $f(x) \to -\infty$ (falls on the right)
• As $x \to -\infty$, $f(x) \to +\infty$ (rises on the left)
• Shape: falling from upper left to lower right

Zeros, Roots, and X-Intercepts

These three terms all refer to the same thing: the input values that make the function equal to zero. If $f(c) = 0$, then:

• $c$ is a zero of the function
• $c$ is a root of the equation $f(x) = 0$
• $(c, 0)$ is an x-intercept of the graph

Finding zeros is one of the most important skills in working with polynomials. When you solve $f(x) = 0$, you are finding where the graph crosses or touches the x-axis.

Multiplicity of Zeros

The multiplicity of a zero tells you how many times that factor appears in the polynomial. It determines how the graph behaves at that x-intercept.

Consider $f(x) = (x - 2)^3(x + 1)^2$.

The zero $x = 2$ has multiplicity 3 (the factor $(x-2)$ appears three times). The zero $x = -1$ has multiplicity 2 (the factor $(x+1)$ appears two times).

Here is the key insight:

Odd multiplicity: The graph crosses the x-axis at this zero. It passes through from one side to the other.

Even multiplicity: The graph touches the x-axis and bounces back. It approaches the axis, touches it, then turns around.

Higher multiplicities create flatter approaches to the x-axis. A zero with multiplicity 1 crosses sharply. A zero with multiplicity 3 still crosses, but with a flatter appearance at the intercept.

The Fundamental Theorem of Algebra

This theorem makes a powerful guarantee: every polynomial of degree $n$ has exactly $n$ zeros when counted with multiplicity (and allowing for complex numbers).

What does this mean practically?

• A degree 3 polynomial has exactly 3 zeros (counted with multiplicity)
• A degree 5 polynomial has exactly 5 zeros (counted with multiplicity)

Some of these zeros might be repeated (multiplicity greater than 1), and some might be complex numbers (involving $i = \sqrt{-1}$), but the total count always equals the degree.

An important consequence: a polynomial of degree $n$ can have at most $n$ real x-intercepts.

Notation and Terminology

Naming Polynomials by Degree

Finding Zeros: Techniques

Factoring

When a polynomial can be factored, setting each factor equal to zero gives you the roots directly.

The Rational Root Theorem

When factoring is not obvious, the Rational Root Theorem helps you find candidates for rational zeros. It states:

If a polynomial $f(x) = a_n x^n + \cdots + a_1 x + a_0$ has a rational zero $\frac{p}{q}$ (in lowest terms), then:

• $p$ is a factor of the constant term $a_0$
• $q$ is a factor of the leading coefficient $a_n$

This gives you a finite list of possible rational zeros to test.

Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a linear factor $(x - c)$. It is faster than long division and helps you test potential zeros and factor polynomials.

1 |  1   -6   11   -6
  |       1   -5    6
  |_________________
     1   -5    6    0

Graphing Polynomial Functions

When graphing a polynomial, follow these steps:

1. Find the degree and leading coefficient to determine end behavior
2. Find the zeros and note their multiplicities
3. Find the y-intercept by calculating $f(0)$
4. Plot additional points if needed for accuracy
5. Connect with a smooth curve (polynomials are always smooth, no sharp corners)

Writing Polynomial Equations

Sometimes you need to work backwards: given information about zeros or points on a graph, write the polynomial function.

Key Properties

Maximum Number of Turning Points

A polynomial of degree $n$ can have at most $n - 1$ turning points (local maxima or minima).

• A linear function (degree 1) has 0 turning points
• A quadratic (degree 2) has at most 1 turning point
• A cubic (degree 3) has at most 2 turning points
• A quartic (degree 4) has at most 3 turning points

Relationship Between Zeros and Factors

If $c$ is a zero of polynomial $f(x)$, then $(x - c)$ is a factor of $f(x)$.

Conversely, if $(x - c)$ is a factor of $f(x)$, then $c$ is a zero of $f(x)$.

Complex Conjugate Pairs

If a polynomial has real coefficients and $a + bi$ is a complex zero (where $b \neq 0$), then its conjugate $a - bi$ is also a zero.

This means complex zeros always come in pairs for polynomials with real coefficients.

Intermediate Value Theorem

If $f$ is a polynomial function and $f(a)$ and $f(b)$ have opposite signs, then there is at least one zero between $a$ and $b$.

This theorem helps locate zeros: if the function is positive at one point and negative at another, it must cross zero somewhere in between.

Real-World Applications

Modeling Physical Phenomena

Polynomial functions model many real-world situations:

Projectile motion: The height of a thrown ball follows a quadratic (degree 2) path: $$h(t) = -16t^2 + v_0 t + h_0$$

Roller coaster design: Higher-degree polynomials can model the ups and downs of a roller coaster track.

Business and Economics

Profit functions: When revenue and cost functions are polynomials, the profit function $P(x) = R(x) - C(x)$ is also a polynomial. Finding where $P(x) = 0$ tells you the break-even points.

Optimization: Finding maximum profit or minimum cost often involves analyzing polynomial functions.

Engineering and Design

Curve fitting: Engineers use polynomials to create smooth curves that pass through specific points. Computer graphics and CAD software rely heavily on polynomial curves (like Bezier curves).

Signal processing: Polynomial functions help model and filter signals in electronics and communications.

Population Models

While exponential models are common for population growth, polynomial models can approximate population changes over shorter time periods or when growth rates change.

Volume Calculations

When you cut squares from the corners of a rectangular sheet to make an open box, the volume is a polynomial function of the square’s side length: $$V(x) = x(L - 2x)(W - 2x)$$

This cubic function helps determine what size squares to cut to maximize the box’s volume.

Self-Test Problems

1 |  1   0  -7   6
  |      1   1  -6
  |_______________
     1   1  -6   0

Summary

• A polynomial function has the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ where all exponents are non-negative integers.

• The degree is the highest exponent; the leading coefficient is the coefficient of the highest-degree term.

• End behavior depends on degree and leading coefficient:

  • Even degree: both ends go the same direction
  • Odd degree: ends go opposite directions
  • Positive leading coefficient: right side goes up
  • Negative leading coefficient: right side goes down

• Zeros, roots, and x-intercepts are different names for values where $f(x) = 0$.

• Multiplicity determines graph behavior at zeros:

  • Odd multiplicity: graph crosses the x-axis
  • Even multiplicity: graph touches and bounces

• The Fundamental Theorem of Algebra guarantees a degree-$n$ polynomial has exactly $n$ zeros (counting multiplicity and complex numbers).

• Finding zeros techniques include:

  • Factoring
  • Rational Root Theorem (possible zeros are $\pm\frac{p}{q}$ where $p$ divides the constant term and $q$ divides the leading coefficient)
  • Synthetic division for testing candidates and factoring

• A degree-$n$ polynomial has at most $n - 1$ turning points.

• To write a polynomial from zeros: create factors $(x - c)$ for each zero $c$, raise each to its multiplicity, multiply by a leading coefficient $a$, and use any additional given points to solve for $a$.

• Polynomials model many real-world phenomena including projectile motion, profit functions, engineering curves, and volume optimization problems.