Degrees of Polynomials 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 of variables. 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 power of the variable in the polynomial. For example, in the polynomial 3x³ + 2x² - 5x + 7, the highest power of x 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 - 6x² + 11x - 6)
Higher-degree polynomials: Degree 4 or higher

Note: The degree of a polynomial is always a non-negative integer. It cannot be negative or a fraction.

How to Calculate Polynomial Degree

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

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

Formula: Degree of polynomial = Highest exponent of the variable in the polynomial.

For example, consider the polynomial 4x⁴ - 3x³ + 2x² - x + 5. The exponents of x in each term are 4, 3, 2, 1, and 0. The highest exponent is 4, so the degree of the polynomial is 4.

Examples

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

Example 1: Linear Polynomial

Polynomial: 5x + 2

Exponents: 1 (for 5x) and 0 (for 2)

Highest exponent: 1

Degree: 1

Example 2: Quadratic Polynomial

Polynomial: x² - 4x + 4

Exponents: 2 (for x²), 1 (for -4x), and 0 (for 4)

Highest exponent: 2

Degree: 2

Example 3: Cubic Polynomial

Polynomial: 2x³ + 3x² - x - 1

Exponents: 3 (for 2x³), 2 (for 3x²), 1 (for -x), and 0 (for -1)

Highest exponent: 3

Degree: 3

FAQ

What is the degree of a constant polynomial?: The degree of a constant polynomial (e.g., 5) is 0 because there is no variable in the polynomial.
Can the degree of a polynomial be negative?: No, the degree of a polynomial is always a non-negative integer. It cannot be negative or a fraction.
How do I determine the degree of a polynomial with multiple variables?: For polynomials with multiple variables, the degree is the highest sum of exponents of all variables in any single term. For example, in the polynomial xy² + x²y, the degree is 3 (from the term xy² where 1+2=3).
What is the difference between the degree of a polynomial and its order?: The terms "degree" and "order" are often used interchangeably in the context of polynomials. Both refer to the highest power of the variable in the polynomial.