Degree of a Polynomial

In this tutorial, we’ll see what is meant by the degree of a polynomial. But before we can dive into that, there are a few things to keep in mind.

Let’s start with an example of a polynomial.

5a^3b^2 + a^2b^2 - 2ab^3 + 4b

So here we have a polynomial with four terms. Terms are the parts of a polynomial separated by addition/subtraction symbols.

Now, a term in a polynomial typically has a coefficient attached to one or more variables with exponents on them.

Degree of a Term

The degree of a term is the sum of the powers (or exponents) of the variables in the term.

For example, consider the following term —

7x^3y^2z^6

The term contains three variables, $\hspace{0.2em} x \hspace{0.2em}$ , $\hspace{0.2em} y \hspace{0.2em}$ , and $\hspace{0.2em} z \hspace{0.2em}$ . And their powers are $\hspace{0.2em} 3 \hspace{0.2em}$ , $\hspace{0.2em} 2 \hspace{0.2em}$ , and $\hspace{0.2em} 6 \hspace{0.2em}$ . So the degree of the term would be their sum, $\hspace{0.2em} 11 \hspace{0.2em}$ .

\begin{align*} \text{deg }(7x^ {\color{Red} 3} y^ {\color{Red} 2} z^ {\color{Red} 6} ) \, &= \, {\color{Red} 3} + {\color{Red} 2} + {\color{Red} 6} \\[1em] &= \, {\color{Red} 11} \end{align*}

And remember, if a variable does not have a visible exponent, its exponent is $\hspace{0.2em} 1 \hspace{0.2em}$ .

Here’s an example.

\begin{align*} \text{deg }(6p^2q) \, &= \, \text{deg }(6p^2q^ {\color{Red} 1} ) \\[1em] &= \, 2 + {\color{Red} 1} \, = \, 3 \end{align*}

What Is the Degree of a Polynomial?

The degree of a polynomial is the highest of the degrees of its terms.

For the polynomial above, the degree of each term is shown below in the blue boxes. And since the highest of the degrees of $\hspace{0.2em} 5 \hspace{0.2em}$ , the degree of the polynomial is $\hspace{0.2em} 5 \hspace{0.2em}$ .

How to Find the Degree of a Polynomial

Here are a couple of examples to demonstrate how to find the degree of a polynomial.

Example

What is the degree of the polynomial below?

x^3y - 6x^4y^5 - 5x^2y^3 + 7x - 2

Solution

There are two simple steps.

Step 1. Add the exponents of the variables in each term to get the degree of each term.

I have written the degree of each term below it.

Step 2. Pick the highest of the degrees. That’s the degree of the polynomial.

Here, the highest degree is $\hspace{0.2em} 9 \hspace{0.2em}$ (for the second term). So the degree of the polynomial would be $\hspace{0.2em} 9 \hspace{0.2em}$ .

Example

What is the degree of the polynomial below?

p^2qr^3 + 2r^7 + 5pq^3 - q^2r^2

Solution

Again, we start by writing the degree of each term below it.

Since $\hspace{0.2em} 7 \hspace{0.2em}$ is the highest among them, that’s the degree of our polynomial.

Polynomials With Only One Variable

Now very often, the polynomials we are working with have one variable only. For example —

-2x^2 + x^3 + 3x^4 - 9x + 1

In such polynomials, the degree of a term is the same as the power of its only variable (there is no need to do any addition). That means the degree of the polynomial would be the highest power of the variable in it.

So, the degree of the polynomial above is $\hspace{0.2em} 4 \hspace{0.2em}$ , the highest power of $\hspace{0.2em} x \hspace{0.2em}$ in it.

Similarly, the degree of the polynomial below is $\hspace{0.2em} 3 \hspace{0.2em}$ .

x^3 + 4x^2 - 1

Degree of a Constant Polynomial

A constant polynomial is a polynomial with only a constant and no variables (apparently). For example, $\hspace{0.2em} 10 \hspace{0.2em}$ .

The degree of a constant polynomial (except if it’s a zero polynomial) is $\hspace{0.2em} 0 \hspace{0.2em}$ . Here’s why.

You can rewrite the constant as the constant times some variable raised to the zeroth power. (Remember, anything to the zeroth power is $\hspace{0.2em} 1 \hspace{0.2em}$ ).

10 \, =\, 10 \cdot {\color{Red} 1} \, =\, 10 \, {\color{Red} x^0}

Now once you write the constant polynomial as that, it becomes clear why its degree is $\hspace{0.2em} 1 \hspace{0.2em}$ .

Degree of a Zero Polynomial

A constant polynomial whose value is $\hspace{0.2em} 0 \hspace{0.2em}$ is known as a zero polynomial.

The degree of a zero polynomial is generally considered undefined. One reason is as follows.

0 = 0 \, {\color{Red} x^n}

As you can see above, the exponent of $\hspace{0.2em} x \hspace{0.2em}$ , $\hspace{0.2em} n \hspace{0.2em}$ , can be anything and it wouldn’t matter. That means the degree of a zero polynomial can be anything.

Special Polynomials Based on Their Degrees

Depending on their degrees, certain polynomials have special names. For example, a polynomial of degree $\hspace{0.2em} 1 \hspace{0.2em}$ is known as a linear polynomial.

The following table lists the names of polynomials with degrees $\hspace{0.2em} 0 \hspace{0.2em}$ through $\hspace{0.2em} 5 \hspace{0.2em}$ .

Type	Degree	Example
Constant Polynomial	$\hspace{0.2em} 0 \hspace{0.2em}$	$\hspace{0.2em} 5 \hspace{0.2em}$
Linear Polynomial	$\hspace{0.2em} 1 \hspace{0.2em}$	$\hspace{0.2em} 3x - 1 \hspace{0.2em}$
Quadratic Polynomial	$\hspace{0.2em} 2 \hspace{0.2em}$	$\hspace{0.2em} x^2 + 4x - 2 \hspace{0.2em}$
Cubic Polynomial	$\hspace{0.2em} 3 \hspace{0.2em}$	$\hspace{0.2em} x^3 + 7x \hspace{0.2em}$
Quartic Polynomial	$\hspace{0.2em} 4 \hspace{0.2em}$	$\hspace{0.2em} x^4 + 5x^2 - 1 \hspace{0.2em}$
Quintic Polynomial	$\hspace{0.2em} 5 \hspace{0.2em}$	$\hspace{0.2em} x^5 \hspace{0.2em}$

Observe how it’s only the degree of the polynomial that matters. Not the number of terms or anything else.

And with that, we come to the end of this tutorial on the degree of a polynomial. Until next time.