Degree Of A Polynomial

What Is the Degree of a Polynomial?

The degree of a polynomial is the highest exponent of the variable in any one of its terms. It's a single number that gives us important information about the polynomial's behavior and classification. To fully grasp this, let's quickly review what a 'term' is.

A polynomial is an expression made up of one or more terms that are added or subtracted. A term is a combination of a number (the coefficient) and one or more variables raised to non-negative integer powers (the exponents).

For example, in the polynomial below:

$$5x^4-2x^3+7x-9$$

The terms are:

• $5x^4$
• $-2x^3$
• $7x$
• $-9$

To find the degree of the entire polynomial, we first find the degree of each individual term. The degree of a term is simply the exponent of its variable. If there are multiple variables, we'll add their exponents (more on that later!).

• The degree of $5x^4$ is $4$.
• The degree of $-2x^3$ is $3$.
• The degree of $7x$ is $1$ (because $x$ is the same as $x^1$).
• The degree of $-9$ is $0$ (because a constant can be written as $-9x^0$, and $x^0=1$).

Now, to find the degree of the polynomial, we just look for the highest degree among all the terms. In this case, the degrees are 4, 3, 1, and 0. The highest value is 4. Therefore, the degree of the polynomial $5x^4-2x^3+7x-9$ is $4$.

How Do You Find the Degree for a Single-Variable Polynomial?

Finding the degree of a polynomial with just one variable (like $x$ or $y$) is a straightforward process. You just need to be systematic. Here is a simple, three-step method to get it right every time.

1. Identify Each Term: Look at the expression and separate it into its individual terms. Remember that terms are separated by addition and subtraction signs.
2. Find the Degree of Each Term: For each term, look at the exponent on the variable. This number is the degree of that specific term. Don't forget that a variable with no visible exponent has a degree of 1 (e.g., $3x=3x^1$) and a constant term has a degree of 0 (e.g., $12=12x^0$).
3. Select the Highest Degree: Compare all the term degrees you found in the previous step. The single largest value is the degree of the entire polynomial.

Let's walk through an example to see this method in action.

Notice that the term with the highest degree, $-x^5$, was not written first. This is why it's crucial to check every term and not just assume the first one gives you the answer!

What About Polynomials with Multiple Variables?

Sometimes you'll encounter polynomials that have more than one variable in a single term, like $x$ and $y$ together. The rule for finding the degree changes slightly, but it's just as easy once you know it.

To find the degree of a term with multiple variables, you add the exponents of all the variables within that term. Let's look at the term $6x^2{y}^5$. To find its degree, we add the exponent of $x$ (which is 2) and the exponent of $y$ (which is 5). So, the degree of this term is $2+5=7$.

Once you find the degree of every term using this addition rule, the process is the same as before: the degree of the entire polynomial is the highest degree you found among all the terms.

What Are the Special Names for Polynomials Based on Degree?

In algebra, you'll hear certain types of polynomials mentioned over and over. These common types are named based on their degree. Knowing these names is essential for understanding the concepts you'll learn in class. It's like learning the difference between a square and a triangle in geometry—they have different properties and are treated differently.

Here is a table of the most common names for polynomials based on their degree:

For polynomials with a degree higher than 5, we typically don't use special names. Instead, we just refer to them by their degree. For example, you would simply call $x^7-2x^3+5$ a "seventh-degree polynomial" or a "polynomial of degree 7."

Why Is the Degree So Important in Algebra?

The degree of a polynomial isn't just a random piece of trivia; it's one of the most important characteristics you can identify. It gives you a roadmap for understanding and working with the polynomial. Here are three key reasons why the degree matters so much:

1. It Predicts the Graph's Shape: The degree gives you a fundamental clue about what the graph of the polynomial will look like.
   • Degree 1 (Linear): The graph is always a straight line.
   • Degree 2 (Quadratic): The graph is always a parabola, which is a 'U' shape that opens either upwards or downwards.
   • Degree 3 (Cubic): The graph typically has an 'S' shape with up to two turns.
   • Higher Degrees: As the degree increases, the graph can have more turns and become more complex. The maximum number of turns in the graph is always one less than the degree ($n-1$).
2. It Determines the Number of Solutions: The degree tells you the maximum number of solutions (also called roots or x-intercepts) a polynomial equation can have. The Fundamental Theorem of Algebra states that a polynomial of degree $n$ has exactly $n$ complex roots. For what you'll see in Algebra 1, this means a polynomial of degree $n$ can cross the x-axis at most $n$ times. For example, a quadratic equation (degree 2) can have at most two real solutions.
3. It Helps Classify and Solve Equations: Knowing the degree tells you what kind of equation you're dealing with and what methods to use to solve it. You use different strategies to solve a linear equation (like $2x-6=0$) than you do to solve a quadratic equation (like $x^2+5x+6=0$), which often requires factoring or the quadratic formula. The degree is the first thing you look at to classify the problem.

What Is Standard Form and How Does It Help?

When you write a polynomial, there's a conventional way to organize it called standard form. Writing a polynomial in standard form means arranging its terms in order from the highest degree to the lowest degree.

For example, the polynomial $4x^2-7+2x^3-5x$ is not in standard form. To put it in standard form, we find the degree of each term (3, 2, 1, and 0) and reorder them from largest to smallest:

$$2x^3+4x^2-5x-7$$

Writing a polynomial in standard form is incredibly useful for two reasons:

1. It makes the degree obvious. The degree of the polynomial is simply the degree of the very first term. No more hunting required!
2. It helps identify the leading coefficient. The leading coefficient is the coefficient of the term with the highest degree. In the example above, the leading coefficient is $2$. The leading coefficient is important because it tells you about the graph's end behavior (what happens to the graph as $x$ goes to positive or negative infinity).

What Are Common Mistakes When Finding the Degree?

Finding the degree is usually simple, but a few common trip-ups can lead to the wrong answer. Be on the lookout for these mistakes!

• Confusing Degree with the Number of Terms: A polynomial like $x^7+2x-1$ has 3 terms (making it a trinomial), but its degree is $7$. The number of terms and the degree are two completely different characteristics.
• Assuming the First Term Gives the Degree: Always scan the entire polynomial! In $5x^2+3x^4-2$, the degree is $4$, not $2$. This mistake happens most often when a polynomial is not written in standard form.
• Forgetting Implied Exponents of 1 and 0: It's easy to overlook a term like $-x$ or a constant like $+8$. Remember that $-x$ has a degree of 1 (it's $-x^1$) and $+8$ has a degree of 0 (it's $8x^0$). These can still be the highest degree if, for example, the polynomial is linear (e.g., $6x-2$, degree 1) or constant (e.g., $15$, degree 0).
• Incorrectly Handling Multiple Variables: For a term like $8x^2{y}^4$, a common mistake is to say the degree is 4 (the highest exponent) or 2. The correct method is to add the exponents: $2+4=6$.
• Mistaking a Non-Polynomial for a Polynomial: An expression is only a polynomial if its variables have non-negative, integer exponents. An expression like $3x^{-2}+5$ or $4\text{[}\sqrt{x}\text{]}-1$ (which is $4x^{1/2}-1$) is not a polynomial, so the concept of degree doesn't apply in the same way.

Quick Summary: Key Concepts

Here's a quick rundown of the most important ideas covered in this lesson. Use this as a reference or a study guide.

• Term: A single part of a polynomial, consisting of a coefficient and a variable raised to a power (e.g., $5x^3$).
• Degree of a Term (Single Variable): The exponent on the variable in that term.
• Degree of a Term (Multiple Variables): The sum of all the exponents on the variables within that single term.
• Degree of a Polynomial: The single highest degree found among all of its terms.
• Standard Form: The conventional way to write a polynomial, by arranging its terms in descending order of degree (from highest to lowest).
• Leading Coefficient: The coefficient of the term with the highest degree, which is the first term when the polynomial is in standard form.
• Naming by Degree: Polynomials are named based on their degree: 0 (Constant), 1 (Linear), 2 (Quadratic), 3 (Cubic), etc.