Table of Contents
Understanding the Degree of a Polynomial Function
In this chapter we focus on two closely related ideas:
- the degree of a polynomial function, and
- how the degree (together with the leading coefficient) affects the overall shape or end behavior of its graph.
We assume you already know what a polynomial function is and how its terms look in general. Here we concentrate on what the highest power tells you about the graph.
Degree and Leading Term
A polynomial function in one variable $x$ can be written (after combining like terms and arranging in descending powers) as:
$$
f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0
$$
where:
- $a_n, a_{n-1}, \dots, a_0$ are real numbers (the coefficients),
- $a_n \neq 0$ (the coefficient of the highest power is not zero),
- $n$ is a nonnegative integer.
The degree of the polynomial is the highest power of $x$ that appears with a nonzero coefficient. In the expression above, the degree is $n$.
The term $a_n x^n$ is called the leading term, and $a_n$ is the leading coefficient.
Identifying the Degree
To find the degree:
- Make sure the polynomial is written as a sum of powers of $x$ (no like terms left uncombined).
- Look for the largest exponent of $x$.
- That exponent is the degree.
Examples:
- $f(x) = 3x^4 - 2x^2 + 7x - 5$
Degree: $ (leading term x^4$, leading coefficient $). - $g(x) = -5x^7 + x^3 - 9$
Degree: $ (leading term $-5x^7$, leading coefficient $-5$). - $h(x) = 6 - 4x + x^2 - x^2$
Combine $x^2 - x^2 = 0$, so $h(x) = 6 - 4x$.
Degree: $ (leading term $-4x$, leading coefficient $-4$). - $k(x) = 8$ (a nonzero constant function)
Degree: $ (can be seen as x^0$).
- The zero polynomial, $f(x) = 0$, is a special case and is usually said to have no degree (or “undefined degree”). We will not focus on this exceptional case here.
End Behavior of Polynomial Functions
The end behavior of a function describes what happens to $f(x)$ as $x$ becomes very large in the positive or negative direction:
- $x \to +\infty$ (“$x$ goes to positive infinity”), and
- $x \to -\infty$ (“$x$ goes to negative infinity”).
For polynomial functions, the leading term dominates the behavior when $|x|$ is very large. That means:
$$
\text{For large } |x|,\quad f(x) \approx a_n x^n.
$$
So, to understand the end behavior of a polynomial, you generally only need to look at:
- the degree $n$, and
- the leading coefficient $a_n$.
Lower-degree terms (like $x^{n-1}$, $x^{n-2}$, …, constants) matter for the detailed shape near the middle of the graph, but they do not change the overall direction of the ends.
Even vs. Odd Degree
The degree $n$ can be even or odd, and this makes a big difference.
Let’s consider the simple “model” functions $x^n$ for various $n$.
Even Degree (like $x^2, x^4, x^6, \dots$)
For $f(x) = x^2$:
- As $x \to +\infty$, $x^2 \to +\infty$.
- As $x \to -\infty$, $x^2 \to +\infty$ as well (because a negative number squared is positive).
So both “ends” of the graph go up.
For higher even powers, like $x^4$ or $x^6$:
- As $x \to +\infty$, $x^n \to +\infty$.
- As $x \to -\infty$, $x^n \to +\infty$.
Again, both ends go up, although the graph may be steeper or flatter in the middle.
General fact:
- If the degree $n$ is even and the leading coefficient $a_n$ is positive, both ends of the graph go up.
- If the degree $n$ is even and the leading coefficient $a_n$ is negative, both ends of the graph go down (because multiplying by a negative flips the graph vertically).
Odd Degree (like $x^1, x^3, x^5, \dots$)
For $f(x) = x$:
- As $x \to +\infty$, $x \to +\infty$.
- As $x \to -\infty$, $x \to -\infty$.
So the right end goes up, and the left end goes down.
For $f(x) = x^3$:
- As $x \to +\infty$, $x^3 \to +\infty$.
- As $x \to -\infty$, $x^3 \to -\infty$.
Again, right end up, left end down.
General fact:
- If the degree $n$ is odd and the leading coefficient $a_n$ is positive, the graph goes:
- down on the left ($x \to -\infty$), and
- up on the right ($x \to +\infty$).
- If the degree $n$ is odd and the leading coefficient $a_n$ is negative, this pattern is reversed:
- up on the left,
- down on the right.
Combining Degree and Leading Coefficient
Putting these facts together, we can summarize all four possibilities for the end behavior of a polynomial.
Let $f(x)$ be a polynomial of degree $n$ with leading term $a_n x^n$.
Case 1: Even Degree, Positive Leading Coefficient
- Degree $n$ is even.
- Leading coefficient $a_n > 0$.
End behavior:
- As $x \to +\infty$, $f(x) \to +\infty$.
- As $x \to -\infty$, $f(x) \to +\infty$.
Both ends go up.
Example:
$$
f(x) = 2x^4 - 5x^2 + 1
$$
The leading term is $2x^4$; degree $4$ (even), $a_4 = 2 > 0$.
Case 2: Even Degree, Negative Leading Coefficient
- Degree $n$ is even.
- Leading coefficient $a_n < 0$.
End behavior:
- As $x \to +\infty$, $f(x) \to -\infty$.
- As $x \to -\infty$, $f(x) \to -\infty$.
Both ends go down.
Example:
$$
g(x) = -3x^6 + x^2 - 7
$$
Leading term $-3x^6$; degree $6$ (even), $a_6 = -3 < 0$.
Case 3: Odd Degree, Positive Leading Coefficient
- Degree $n$ is odd.
- Leading coefficient $a_n > 0$.
End behavior:
- As $x \to -\infty$, $f(x) \to -\infty$.
- As $x \to +\infty$, $f(x) \to +\infty$.
Left end down, right end up.
Example:
$$
h(x) = x^3 - 4x + 2
$$
Leading term $x^3$; degree $3$ (odd), $a_3 = 1 > 0$.
Case 4: Odd Degree, Negative Leading Coefficient
- Degree $n$ is odd.
- Leading coefficient $a_n < 0$.
End behavior:
- As $x \to -\infty$, $f(x) \to +\infty$.
- As $x \to +\infty$, $f(x) \to -\infty$.
Left end up, right end down.
Example:
$$
p(x) = -5x^5 + x^2 - 1
$$
Leading term $-5x^5$; degree $5$ (odd), $a_5 = -5 < 0$.
Visual Patterns to Remember
You do not need to memorize every individual example if you remember the basic patterns. Think of a very large $|x|$ and focus on the leading term.
You can summarize the shapes like this (where “up” means $f(x) \to +\infty$ and “down” means $f(x) \to -\infty$):
- Even degree ($n$ even)
- $a_n > 0$: up on the left, up on the right (like a “U” shape at the ends).
- $a_n < 0$: down on the left, down on the right (upside-down “U” at the ends).
- Odd degree ($n$ odd)
- $a_n > 0$: down on the left, up on the right.
- $a_n < 0$: up on the left, down on the right.
These patterns do not tell you everything about the graph (such as where it crosses the $x$-axis or how many turning points it has), but they always hold for polynomial functions and give you a reliable sketch of how the ends behave.
Using End Behavior in Practice
When you study polynomial functions further (for example, when analyzing zeros or graphing in more detail), you will often start by:
- Finding the degree and leading coefficient.
- Using them to determine the end behavior.
- Combining this with other information (like zeros, multiplicities, and values at certain points) to sketch or understand the whole graph.
For now, make sure you can:
- Identify the degree from a given polynomial expression.
- Identify the leading coefficient.
- Decide, from those two pieces of information, what happens to $f(x)$ as $x \to +\infty$ and as $x \to -\infty$.