Polynomial Degrees Calculator

A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest power of the variable in the polynomial.

What is a Polynomial Degree?

The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, in the polynomial 3x³ + 2x² - 5x + 7, the highest exponent is 3, so the degree of the polynomial is 3.

Polynomials can be classified based on their degrees:

Constant polynomial: Degree 0 (e.g., 5)
Linear polynomial: Degree 1 (e.g., 2x + 3)
Quadratic polynomial: Degree 2 (e.g., x² - 4x + 4)
Cubic polynomial: Degree 3 (e.g., x³ + 2x² - 5)
Higher-degree polynomials: Degree 4 or higher

Note: The degree of a polynomial is always a non-negative integer. It is not possible to have a negative or fractional degree in a polynomial.

How to Calculate Polynomial Degree

To calculate the degree of a polynomial, follow these steps:

Identify all the terms in the polynomial.
Determine the exponent of the variable in each term.
Find the highest exponent among all the terms.
The highest exponent is the degree of the polynomial.

Formula: Degree of polynomial P(x) = max(exponent of each term in P(x))

For example, consider the polynomial P(x) = 4x⁵ - 3x³ + 2x² - x + 7. The exponents of the terms are 5, 3, 2, 1, and 0. The highest exponent is 5, so the degree of the polynomial is 5.

Examples

Let's look at a few examples to understand how to calculate the degree of a polynomial.

Example 1: Linear Polynomial

Consider the polynomial P(x) = 2x + 3.

The exponents of the terms are 1 and 0. The highest exponent is 1, so the degree of the polynomial is 1.

Example 2: Quadratic Polynomial

Consider the polynomial P(x) = x² - 4x + 4.

The exponents of the terms are 2, 1, and 0. The highest exponent is 2, so the degree of the polynomial is 2.

Example 3: Cubic Polynomial

Consider the polynomial P(x) = x³ + 2x² - 5.

The exponents of the terms are 3, 2, and 0. The highest exponent is 3, so the degree of the polynomial is 3.

FAQ

What is the degree of a constant polynomial?: The degree of a constant polynomial is 0 because there are no variables in the polynomial.
Can a polynomial have a negative degree?: No, the degree of a polynomial is always a non-negative integer. It is not possible to have a negative degree.
What is the degree of the zero polynomial?: The zero polynomial is considered to have an undefined degree. However, in some contexts, it is assigned a degree of -∞.
How do I find the degree of a polynomial with multiple variables?: For a polynomial with multiple variables, the degree is the highest sum of the exponents of the variables in any single term.