Find The Polynomial Degree Calculator

Calculate the Degree of a Polynomial

What is the Degree of a Polynomial?

The degree of a polynomial is the highest power (exponent) of its variable(s) found in any of its terms, after the polynomial has been expanded and simplified. It's a fundamental concept in algebra that helps classify polynomials and understand their behavior.

For a polynomial in a single variable (like 'x'), the degree is simply the largest exponent of 'x'. For example, in the polynomial 5x^3 - 2x + 7, the terms are 5x^3, -2x (which is -2x^1), and 7 (which is 7x^0). The exponents are 3, 1, and 0. The highest exponent is 3, so the degree of the polynomial is 3.

If a polynomial has multiple variables in a single term (e.g., 3x^2y^3), the degree of that term is the sum of the exponents of the variables (2 + 3 = 5). The degree of the polynomial is then the largest sum found across all terms. However, our calculator focuses on single-variable polynomials or those where variables are not multiplied within a term for simplicity, looking for the highest exponent of 'x'.

Anyone studying algebra, calculus, or any field that uses mathematical modeling will need to understand and find the degree of a polynomial. It's used to determine the number of roots a polynomial might have, its end behavior, and in various other mathematical analyses.

A common misconception is that the number of terms determines the degree. The number of terms and the degree are different concepts. For instance, x^5 + 1 has a degree of 5 but only two terms, while x^2 + x + 1 has a degree of 2 but three terms.

Polynomial Degree Formula and Mathematical Explanation

To find the degree of a polynomial in a single variable (let's say 'x'), you look at each term and find the exponent of 'x'.

A term in a polynomial generally looks like ax^n, where 'a' is the coefficient and 'n' is the exponent of 'x'.

1. Identify the terms: Separate the polynomial into individual terms based on the '+' and '-' signs. Remember that a minus sign goes with the term it precedes.
2. For each term, find the exponent of the variable:
   • If the term is like ax^n, the exponent is 'n'.
   • If the term is like ax, it's ax^1, so the exponent is 1.
   • If the term is a constant 'a', it's ax^0, so the exponent is 0.
3. Find the highest exponent: The largest exponent found among all terms is the degree of the polynomial.

For example, for P(x) = 4x^5 - 7x^2 + x - 9:

• Term 4x^5: exponent is 5.
• Term -7x^2: exponent is 2.
• Term x (or 1x^1): exponent is 1.
• Term -9 (or -9x^0): exponent is 0.

The highest exponent is 5, so the degree of the polynomial is 5.

Variables in Polynomial Terms

Variable/Component	Meaning	Example in 4x^5
a (coefficient)	The numerical multiplier of the variable part.	4
x (variable)	The base variable.	x
n (exponent/power)	The power to which the variable is raised.	5
Term degree	The exponent 'n' in a single variable term.	5

Practical Examples (Real-World Use Cases)

Let's find the degree of a polynomial for a couple of examples.

Example 1: 6x^3 - 2x^5 + 3x - 1

• Terms: 6x^3, -2x^5, 3x, -1
• Exponents: 3 (from x^3), 5 (from x^5), 1 (from x), 0 (from -1)
• Highest exponent: 5
• Degree of the polynomial: 5

Example 2: y^2 + 8y^7 - 5y^3 + 2 (using y as variable)

• Terms: y^2, 8y^7, -5y^3, 2
• Exponents: 2, 7, 3, 0
• Highest exponent: 7
• Degree of the polynomial: 7 (assuming single variable y)

Example 3: 10 - x

• Terms: 10, -x (or -1x^1)
• Exponents: 0, 1
• Highest exponent: 1
• Degree of the polynomial: 1 (This is a linear polynomial)

Example 4: 42

• Term: 42 (or 42x^0)
• Exponent: 0
• Highest exponent: 0
• Degree of the polynomial: 0 (This is a constant polynomial)

How to Use This Polynomial Degree Calculator

1. Enter the Polynomial: Type or paste your polynomial into the input field labeled "Enter Polynomial". Use 'x' as the variable (e.g., 5x^3 - x + 2). You can use other variables like 'y' or 'z', but the calculator is primarily set up to look for 'x' first. If 'x' is not found, it will look for 'y', then 'z', and report the highest power of the first one it finds. For consistent results, stick to 'x'.
2. Exponents: Use the caret symbol '^' to denote exponents (e.g., x^4 for x to the power of 4). If a variable has no exponent, it is assumed to be 1 (e.g., 3x is 3x^1).
3. Terms: Separate terms using '+' or '-'. Make sure there are spaces or signs between terms.
4. Calculate: Click the "Calculate Degree" button or simply type in the input field. The results will update automatically.
5. View Results: The calculator will display the calculated degree of the polynomial in the highlighted "Primary Result" section. It will also show the polynomial you entered and a table of terms with their individual degrees, plus a chart of coefficients if applicable.
6. Reset: Click "Reset" to clear the input and results and go back to the default example.

The degree of a polynomial is a key piece of information. A degree 'n' polynomial can have up to 'n' real roots (x-intercepts) and 'n-1' turning points (local maxima or minima). Knowing the degree helps in sketching the graph and understanding the polynomial's behavior as x approaches infinity or negative infinity.

Key Factors That Affect Polynomial Degree Results

The calculated degree of a polynomial depends entirely on the exponents of the variables in its terms.

1. Highest Exponent Present: The degree is directly determined by the largest exponent of the variable 'x' (or the primary variable) in any term.
2. Presence of the Variable: If the variable 'x' appears in the polynomial, the degree will be at least 1, unless it only appears as x^0 (in constant terms). If 'x' is not present at all (it's a constant polynomial like "7"), the degree is 0.
3. Terms with Zero Coefficients: Even if a term like 0x^5 is technically present, it equals zero and doesn't contribute to the polynomial in its simplified form, so it wouldn't be used to determine the degree unless it was the only term indicating a higher power before simplification. Our calculator processes the input as given. If you input 0x^5 + 2x^2, it will find 5 as the highest power from the input.
4. Simplification State: The degree is defined for the expanded and simplified form. If you input (x^2+1)(x^3+1), it's a 5th-degree polynomial after expansion, but our simple calculator won't expand it; it would look at x^2 and x^3 as they appear if entered that way (which it doesn't parse well). You should enter the expanded form like x^5 + x^3 + x^2 + 1 to get the correct degree of 5.
5. Single vs. Multiple Variables: Our calculator is primarily designed for single-variable polynomials or those where variables are not multiplied within a term (e.g., it looks for the highest power of 'x', then 'y', then 'z' if 'x' isn't found). For true multivariate polynomials like x^2y^3 + z^4, the degree of a term is the sum of exponents (5 and 4 here), and the polynomial's degree is the max of these (5). Our calculator is simplified and would report 3 for y if x wasn't present, or 4 for z if x and y weren't. We focus on the highest exponent of 'x'.
6. Constants: A non-zero constant term (like '5' or '-3') is considered to have a degree of 0 because it can be written as 5x^0 or -3x^0. The polynomial '0' is sometimes considered to have a degree of -1 or -infinity, but our calculator treats '0' as having degree 0 if entered alone.

Frequently Asked Questions (FAQ)

What is the degree of a constant polynomial like 7?

The degree of a polynomial like 7 (or any non-zero constant) is 0, because 7 can be written as 7x^0.

What is the degree of the zero polynomial (0)?

By convention, the degree of the zero polynomial (0) is often defined as -1 or -∞ (negative infinity) because it has no non-zero terms from which to get a highest power. However, our calculator, if given "0", will likely interpret it as 0x^0 and report degree 0.

Can the degree of a polynomial be negative or a fraction?

For standard polynomials, the exponents must be non-negative integers (0, 1, 2, …). So, the degree of a polynomial is always a non-negative integer (or undefined/-1/-∞ for the zero polynomial). Expressions with negative or fractional exponents are not considered polynomials in the strict sense (they might be rational functions or other types of expressions).

What if my polynomial has more than one variable, like x^2y + y^3?

Our calculator is primarily set to find the highest power of 'x'. If 'x' is not present, it looks for 'y', then 'z'. For x^2y + y^3, if it parsed 'y' in the first term as base, it's complex. If we treat x as the main variable, the first term involves x^2. If y, then y^1 and y^3. It will likely report 3 based on y^3 if x is not found as the primary search. For true multivariate degree, you sum exponents in each term (2+1=3, 3) and take the max (3).

Does the coefficient affect the degree?

No, the coefficients (the numbers multiplying the variables) do not affect the degree of a polynomial, as long as the coefficient of the highest power term is not zero.

What is a linear polynomial?

A linear polynomial is a polynomial of degree 1, like 2x + 5.

What is a quadratic polynomial?

A quadratic polynomial is a polynomial of degree 2, like 3x^2 - x + 4.

What is a cubic polynomial?

A cubic polynomial is a polynomial of degree 3, like x^3 - 7x.