Degree of a Polynomial: Definition, Types, Examples, Facts, FAQs

What Is the Degree of a Polynomial?

The degree of a polynomial is the highest degree among the degrees of the individual terms present in the polynomial.

• If the polynomial is in a single variable, the degree of a polynomial is the highest exponent of the variable with a non-zero coefficient. 

• If the polynomial has more than one variable, then we calculate the degree of each term separately and the highest degree among them represents the degree of the given polynomial.

The standard form of degree polynomial is given by 

$p(x) = a_{n}x^{n} + a_{n\;–\;1}x^{n\;–\;1} + a_{n\;–\;2}x^{n\;–\;2} + … + a_{1}x^{1} + a_{0}$

where, $a_{n} \neq 0$. 

Here, the degree of the polynomial p(x) is n, because “n” is the highest power of variable “x.”

We can represent the degree of a polynomial p(x) by Deg(p(x)).

Example: $p(x) = 9x^{4} + 2x^{3} \;–\; 5x^{2} + 4x + 1$

The degree of the polynomial is 4 because the highest power of the variable x is 4.

Degree of a Polynomial: Definition

The degree of a polynomial is the highest power of the variable in the polynomial expression with a non-zero coefficient. The degree of a polynomial can also be defined as the greatest number among the degrees of the individual terms (monomials) with non-zero coefficients. 

Degree of a Polynomial with One Variable

For a polynomial in one variable, the highest power of the variable in a polynomial is the degree of the polynomial. It is the highest exponent value of the variable in the given polynomial.

Examples: 

• Deg $(x^{5} + 4x^{2} + 3x) = 5$
• Deg $(x) = 1$
• Deg $(x^{3} + 2x + 1) = 3$

Degree of a Polynomial with More than One Variable

When finding the degree of a polynomial when a polynomial has more than one variable, we first find the degree of each individual term by adding the exponents of each variable present in the term. 

In other words, we find the degree of each monomial present in the polynomial. Finally, the degree of the polynomial is the largest degree among the degrees of individual terms.

Example: $5x^{5} + 4xy^{2} + 3xy$

• The degree of the term $5x^{5}$ is 5.
• The degree of $4xy^{2}$ is 3 (Sum of exponents $= 1 + 2 = 3)$.
• The term 3xy has a degree of 2 (Sum of exponents $= 1 + 1 = 2)$.

The highest degree is 5. Thus, the degree of polynomial is 5.

How to Find the Degree of a Polynomial

Before finding the degree, first combine all the like terms (terms having the same variables and the same exponents). This way we ensure that no two terms have the same degree. 

Finding Degree of a Polynomial with Only One Variable

Step 1: Write the polynomial expression in standard form. Standard form of a polynomial refers to arranging the terms in descending order of their degrees.

Step 2: Identify the term with the highest power of the variable. This term must have a non-zero coefficient. In the standard form, the highest power term is the leading term.

Step 3: The degree of the polynomial is equal to the degree of the highest power term.

Example 1: $7x^{3} + 5x^{2} \;–\; 3x \;–\; 4x^{3}  + 2 \;–\; 3x^{3}$

Adding the like terms together, we get

$p(x) = (7x^{3} \;–\; 4x^{3} \;–\; 3x^{3})+ 5x^{2} \;–\; 3x + 2$. 

$p(x) = 5x^{2} \;–\; 3x + 2$ …the term $x^{3}$ vanishes due to the 0 coefficient.

This expression is already in standard form.

Here, the term with the highest power of x is $5x^{2}$.

Hence, the degree of the polynomial is 2.

Example 2: $6x^{2} + 5x^{4} \;–\; 3x^{3} \;–\; x + 4$

Rearrange the terms in descending order of their degrees.

$x^{4} \;–\; 3x^{3} + 6x^{2} \;–\; x + 4$

Here, the term with the highest power of x is $x^{4}$.

Hence, the degree of the polynomial is 4.

Finding Degree of a Polynomial with More than One Variable

Step 1: Identify each term.

Step 2: Find the degree of each term. To find the degree of a term, add the exponents of variables present.

Step 3: Compare the degrees of individual terms. The highest degree among them is the degree of the polynomial.

Example: $ab^{6}\;−\; a^{4}b^{8} + ab$

Degree of $ab^{6} = 1 + 6 = 7$

Degree of $a^{4}\;b^{8} = 4 + 8 = 12$

Degree of $ab = 1 + 1 = 2$

Highest degree $= 12$

Degree of the given polynomial $= 12$

Degree of Zero Polynomial

The zero polynomial is usually denoted by 0 or by the expression $p(x) = 0$. The zero polynomial has no non-zero terms. A polynomial with zero coefficients is called a zero polynomial. It has no terms with non-zero coefficients.

The degree of a zero polynomial is considered to be undefined. Why so? We can rewrite $p(x) = 0$ as $p(x) = 0. x^{n}$. 

$p(x) = 0. x^{1}$

$p(x) = 0. x^{2}$

$p(x) = 0. x^{3}$

…

So, the degree of a zero polynomial is not defined. It has no degree. 

Note: In mathematical practice, sometimes the degree of the zero polynomial is taken to be −∞ and sometimes it is considered undefined.

Degree of Constant Polynomial

A constant polynomial has no variable term. It only has a constant term. Thus, the degree of a constant polynomial is 0.

The constant polynomial $p(x) = c$ where c is a non-zero constant has degree 0. It can be written as $p(x) = kx^{0}$.

This is because there is no variable term present, and the highest power of x is 0, and $x^{0} = 1$. 

Degree of a Polynomial: Applications

The concept of the degree of a polynomial has important applications in mathematics, science, and engineering. A few examples of how the degree of a polynomial can be used are listed below:

• To figure out the maximum possible roots or solutions a function can have.
• To figure out how many times a function crosses the x-axis on a graph.
• We find the degree of each term to see if the polynomial expression is homogeneous. For example, in the polynomial $6x^{3} + 12xy^{2} + 2y^{3}$, all of the terms have a degree of 3. So it is a degree 3 homogeneous polynomial.

Classification of Polynomials Based on Degree

A specific name has been given to each of the polynomials in accordance with their degree. Let’s classify polynomials according to their degree along with their example.

Degree	Name of Polynomials	Examples
0	Constant Polynomial	$p(x) = 7$
1	Linear Polynomial	$p(x) = x + 2$
2	Quadratic Polynomial	$p(x) = x^{2} \;–\; 3x + 2$
3	Cubic Polynomial	$p(x) = 5x^{3} + 6x^{2} + 3x + 2$
4	Bi-quadratic Polynomial	$p(x) = x^{4} + 2x^{3} + x^{2} \;–\; 1$

Facts about Degree of a Polynomial

Conclusion

In this article, we learned about the degree of polynomial and the method to find the degree of the polynomial. Let’s solve a few examples and practice problems.

Solved Examples on Degree of a Polynomial

1. Determine the degree of the polynomial $p(x) = 10x^{4} + 8x^{2} \;–\; 15x + 18$.

Solution:

$p(x) = 10x^{4} + 8x^{2} \;–\; 15x + 18$

The polynomial is written in the standard form.

In this case, the highest power of x is 4.

Thus, the degree of p(x) is 4.

2. Find the degree and leading coefficient of the polynomial $g(x) = 9x^{5} + 5x^{3} + 7x \;–\; 1$.

Solution: 

$p(x) = 9x^{5} + 5x^{3} + 7x \;–\; 1$

In this case, the highest power of x is 5. 

Thus, the degree of g(x) is 5. 

Leading term $= 9x^{5}$

Leading coefficient $= 9$

3. Find the degree of the polynomial $p(x) =3x^{3}y \;–\; 2x^{2} + 7x^{2}y^{3} \;–\; 99$.

Solution: 

$p(x) =3x^{3}y \;–\; 2x^{2} + 7x^{2}y^{3} \;–\; 99$

The polynomial has more than 2 variables.

In this case, we first find the degree of each term by adding the exponents.

Degree of $3x^{3}y = 3 + 1 = 4$

Degree of $2x^{2} = 2$

Degree of $7x^{2}y^{3} = 2 + 3 = 5$

Degree of $99 = 0$

Highest degree $= 5$

Thus, the degree of $p(x) =3x^{3}y \;–\; 2x^{2} + 7x^{2}y^{3} \;–\; 99$ is 5.

4. The length and width of a rectangular garden are $(2x + 3)$ and $(x \;–\; 2)$ respectively. What is the degree of the polynomial that represents the area of the garden?

Solution:

The area of the garden is given by the product of the length and the width.

Therefore, area of rectangular garden $= (2x + 2)(x \;–\; 2)$

$A(x) = 2x^{2} \;–\; 4x + 2x \;–\; 4$

$A(x) = 2x^{2} \;–\; 2x \;–\; 4$

Hence, the degree of the polynomial that represents the area of the garden is 2.

5. What is the degree of the polynomial $P(x, y) = 5x^{2}y^{2} \;–\; 3xy^{2} + 5x \;–\; 2y$?

Solution:

In a polynomial with more than one variable, the degree can be determined by adding the exponents of each variable.

$P(x, y) = 5x^{2}y^{2} \;–\; 3xy^{2} + 5x \;–\; 2y$

The term with the highest degree is $5x^{2}\;y^{2}$. 

Degree of $5x^{2}\;y^{2} = 2 + 2 = 4$.

Therefore, the degree of the polynomial P(x, y) is 4.

Practice Problems on Degree of a Polynomial

Frequently Asked Questions about the Degree of a Polynomial