Algebra • Polynomials

Degree of a Polynomial Calculator

Determine the degree, leading term, leading coefficient, and algebraic classification of single-variable, multivariable, and factored polynomials with step-by-step exponent analysis.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Accurate: Abstract Algebra & Ring Theory

Interactive Calculator

Polynomial Degree & Term Analyzer

Presets:

Enter Polynomial Expression (single-variable, multivariable, or factored)

Supports powers using caret (x^3), multivariable terms (x^2y^4), and products ((x+1)(x-2)).

Degree of Polynomial

Cubic Polynomial

Classification

Cubic

4 Terms (Polynomial)

Leading Term

4x³

Coefficient: 4

Constant Term

-12

Degree 0 Component

Step-by-Step Degree Derivation

Formal Analytical Steps

Individual Term Breakdown

Ordered by Term Sequence

Term	Coefficient	Variable(s) & Exponents	Term Degree	Status

Direct Answer & Overview

Verified Educational Guide

How to Find the Degree of a Polynomial

The degree of a polynomial is the highest variable exponent among all non-zero terms in the expression. For multivariable terms, the term degree is the sum of the exponents of all variables within that term. For factored polynomials, the degree is the sum of the degrees of the individual factors: deg(P · Q) = deg(P) + deg(Q). Non-zero constants have degree 0, while the zero polynomial P(x) = 0 has an undefined degree (or -∞).

Primary Mathematical Formula Maximum Exponent & Multi-Index Sum Rule

Standard Equation

ƒ(x)

Q.E.D.

\deg(P) = \max_{i} \{ \deg(T_i) \} = \max_{i} \left\{ \sum_{j=1}^{k} \alpha_{i,j} \right\}

For a single variable: highest power n where a_n ≠ 0. For factored expressions: deg(P^m · Q^n) = m·deg(P) + n·deg(Q).

Exact Formula

Input Parameters

Required

Polynomial Expression P(x): Single-variable, multivariable, or factored polynomial.

Variable Power Syntax: Formatted using caret notation (e.g., x^3, y^2).

Expected Outputs

Calculated

Degree: Highest variable exponent or multi-index sum.

Classification: Named by degree (Linear, Quadratic, Cubic, etc.) and term count (Monomial, Binomial, Trinomial).

Leading Term & Coefficient: Term corresponding to the highest degree and its numerical scalar.

Worked Numerical Example

Instant Verification

Find the degree of P(x) = 5x⁴ - 2x⁷ + 8x² - 14

→ Inspect term exponents: 4 (from 5x⁴), 7 (from -2x⁷), 2 (from 8x²), and 0 (from -14). The maximum exponent is 7.

Degree = 7 (Septic Polynomial, Leading Term = -2x⁷)

What Is the Degree of a Polynomial?

In algebra, a polynomial is a mathematical expression composed of variables, coefficients, and exponents combined using solely addition, subtraction, and multiplication. The degree of a polynomial measures its highest order of complexity and directly determines its asymptotic growth, maximum number of real roots, and graphical shape.

For a standard single-variable polynomial in x:

P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ (where a_n ≠ 0)

The non-negative integer n is the degree of the polynomial, denoted as °(P) or deg(P). The term containing this highest power (a_nxⁿ) is called the leading term, and its scalar multiplier a_n is the leading coefficient.

Fundamental Theorem of Algebra

A polynomial of degree n has exactly n complex roots (counting multiplicity). Knowing the degree instantly reveals how many solutions exist when solving P(x) = 0.

End-Behavior Dominance

As x → ±∞, the leading term a_nxⁿ dwarfs all lower-degree terms, completely dictating whether the graph ascends or descends towards infinity.

Single-Variable vs. Multivariable Polynomials

While single-variable polynomials require only locating the highest individual exponent, multivariable polynomials involve terms with multiple distinct variables multiplied together.

The Multivariable Term Degree Rule

The degree of a monomial term c · x^ay^bz^c is the sum of all variable exponents: Degree = a + b + c.

Example Expression	Term Analysis	Calculated Degree	Leading Term
7x⁴ - 3x² + 5	Exponents: 4, 2, 0	4	7x⁴
3x²y³ - 5x⁴y + 2xy - 8	3x²y³ (2+3=5), -5x⁴y (4+1=5), 2xy (1+1=2)	5	3x²y³ (tie with -5x⁴y)
2a³b⁴c² - 9ab⁶	2a³b⁴c² (3+4+2=9), -9ab⁶ (1+6=7)	9	2a³b⁴c²

Degree Theorems: Products, Powers, and Sums

When combining polynomials through algebraic operations, their degrees follow strict theoretical theorems. These laws allow you to find the degree of factored or compounded expressions without expanding large algebraic terms:

Degree Product Theorem

°(P × Q) = °(P) + °(Q)

When multiplying polynomials, exponents add. For example: (x² + 3)(x³ - 2x) ⇒ 2 + 3 = 5. You can factor using our polynomial factoring calculator.

Degree Power Theorem

°(P^k) = k × °(P)

Raising a polynomial of degree d to positive integer power k multiplies the degree. For example: (2x³ - 5)⁴ ⇒ 4 × 3 = 12.

Degree Addition Inequality

°(P + Q) ≤ max(°(P), °(Q))

The degree of a sum never exceeds the maximum degree. If leading terms cancel out (e.g. (x³ + 2) + (-x³ + 4x) = 4x + 2), the degree can drop strictly lower.

Composition Theorem

°(P ∘ Q) = °(P) × °(Q)

In function composition P(Q(x)), the degrees multiply. Test composite functions on our composite function evaluator.

Classification Taxonomy (By Degree and Terms)

Mathematicians classify polynomials using a dual naming system: one name based on the numerical degree, and another based on the number of non-zero terms.

Degree	Formal Name	Standard Form	Dedicated Solver Link
0	Constant	f(x) = c	Horizontal Line
1	Linear	f(x) = mx + b	Slope Calculator
2	Quadratic	f(x) = ax² + bx + c	Quadratic Formula Solver
3	Cubic	f(x) = ax³ + bx² + cx + d	Cubic Equation Solver
4	Quartic (Biquadratic)	f(x) = ax⁴ + bx³ + …	Synthetic Division
5	Quintic	f(x) = ax⁵ + …	Abel-Ruffini Insoluble

1 Term

Monomial

e.g. 7x³

2 Terms

Binomial

e.g. 4x² - 9

3 Terms

Trinomial

e.g. x² - 5x + 6

Step-by-Step Algorithm to Find the Degree

Follow this 4-step procedural algorithm to determine the degree of any algebraic polynomial without error:

Simplify and Combine Like Terms: Expand all products or combine terms sharing identical variable components. If leading opposite terms cancel (e.g. 3x⁴ - 3x⁴), eliminate them before proceeding.

Verify Non-Negative Integer Exponents: Ensure every variable exponent belongs to {0, 1, 2, 3, …}. Check that no variables appear in denominators or under square root symbols.

Calculate Each Term's Degree: For single variables, read the exponent. For multivariable terms, add all exponents together (e.g. x³y² ⇒ 3 + 2 = 5). Constant terms have degree 0.

Identify the Global Maximum: The maximum degree among all non-zero terms is the degree of the polynomial. The corresponding term is your leading term.

Special Cases: Constants, Zero, and Non-Polynomials

Standard textbook questions frequently test edge cases that confuse students:

The Non-Zero Constant: Degree = 0

Any non-zero real number such as P(x) = 42 or P(x) = -π has a degree of 0. This is because 42 = 42x⁰ (since x⁰ = 1 for x ≠ 0). The graph is a flat horizontal line with slope 0.

The Zero Polynomial P(x) = 0: Degree = Undefined (or -∞)

Why isn't P(x) = 0 degree 0? Because 0 = 0x⁰ = 0x¹ = 0x¹⁰⁰. It has no non-zero leading coefficient. Defining °(0) = -∞ preserves the product theorem: °(0 × P) = -∞ + °(P) = -∞ = °(0).

Non-Polynomial Expressions

The following are not polynomials and therefore have no valid polynomial degree:

f(x) = 3x⁻² + 5 (Negative exponent ⇒ rational fraction 3/x² + 5)
f(x) = √x - 4 (Fractional exponent x^1/2 ⇒ radical function)
f(x) = 2^x + 3 (Variable in exponent ⇒ exponential function)
f(x) = 1 / (x² + 1) (Variable in denominator ⇒ rational function)

Step-by-Step Graded Worked Examples

Example 1 • Single-Variable with Unordered Powers Basic Tier

Determine the degree of P(x) = 7 + 2x³ - 5x⁶ + 4x

1. Identify each term: 7 (deg 0), 2x³ (deg 3), -5x⁶ (deg 6), 4x (deg 1)

2. Arrange in descending order (Standard Form): -5x⁶ + 2x³ + 4x + 7

3. Locate highest exponent: 6

Degree = 6 (Sextic Polynomial) | Leading Term = -5x⁶ | Leading Coefficient = -5

Example 2 • Multivariable Exponent Sum Intermediate Tier

Find the degree of Q(x, y) = 4x³y² - 8x⁵y + 3xy⁴ - 12

1. Term 4x³y²: degree = 3 + 2 = 5

2. Term -8x⁵y: degree = 5 + 1 = 6

3. Term 3xy⁴: degree = 1 + 4 = 5

4. Term -12: degree = 0

5. Maximum sum = max(5, 6, 5, 0) = 6

Degree = 6 | Leading Term = -8x⁵y | Classification = 4-Term Polynomial

Example 3 • Factored Form Without Full Expansion Advanced Tier

Find the degree of H(x) = (3x² - 1)³ · (x⁴ + 5x - 2)²

1. Factor 1: (3x² - 1) has base degree 2. Raised to power 3 ⇒ 2 × 3 = 6

2. Factor 2: (x⁴ + 5x - 2) has base degree 4. Raised to power 2 ⇒ 4 × 2 = 8

3. Apply Degree Product Law: deg(Factor 1) + deg(Factor 2) = 6 + 8 = 14

Degree = 14 | Method: deg(P³ · Q²) = 3(2) + 2(4) = 14

Common Pitfalls and Reversal Errors

Picking the First Term Instead of Highest Power

In polynomials not written in standard form like 4 + 3x - 8x⁵, students mistakenly assume the degree is 0 or 1 because 4 or 3x appears first. Always scan the entire expression for the maximum exponent.

Ignoring Invisible Exponents on Single Variables

A variable with no written exponent like 7x has an implicit power of 1 (x¹), NOT 0. In multivariable terms like x²y, the power of y is 1, yielding a term degree of 2 + 1 = 3.

Multiplying Exponents Instead of Adding in Factored Form

When multiplying factors (x³ + 1)(x² - 4), students incorrectly calculate 3 × 2 = 6. Powers add when multiplying like bases (x³ × x² = x⁵), so the degree is 3 + 2 = 5.

Confusing Coefficient with Degree

In 100x², the coefficient is 100, but the degree is strictly 2. The degree relates only to the variable's exponent, regardless of how large or small the numerical coefficient is.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

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Frequently Asked Questions

What is the degree of a polynomial?

The degree of a single-variable polynomial is the highest non-negative exponent of the variable with a non-zero coefficient. For example, in P(x) = 4x³ - 5x² + 7, the highest power of x is 3, making it a cubic polynomial of degree 3.

How do you find the degree of a multivariable polynomial?

In a multivariable polynomial, the degree of each individual term is calculated by summing the exponents of all variables in that term. The overall degree of the polynomial is the maximum of these term degrees. For example, the term 4x²y³ has a term degree of 2 + 3 = 5.

What is the degree of a non-zero constant term like 7 or -12?

A non-zero constant has a degree of 0 because it can be rewritten as c · x⁰ (where x⁰ = 1 for any x ≠ 0). Hence, any standalone non-zero number is a degree 0 constant polynomial.

What is the degree of the zero polynomial P(x) = 0?

In abstract algebra and ring theory, the degree of the zero polynomial P(x) = 0 is formally defined as undefined or -∞ (negative infinity). This definition ensures that the fundamental degree law deg(P · Q) = deg(P) + deg(Q) remains valid when multiplying by zero.

How do you find the degree of a factored polynomial like (x + 2)(x² - 5)?

By the Degree Product Theorem, the degree of a product of polynomials equals the sum of the degrees of its individual factors: deg(P · Q) = deg(P) + deg(Q). For (x + 2)(x² - 5), deg(x + 2) = 1 and deg(x² - 5) = 2, so the overall degree is 1 + 2 = 3 without needing to fully expand the expression.

Can a polynomial have a negative or fractional degree?

No. By definition, a polynomial expression cannot contain negative exponents, fractional exponents, or variables in denominators or under radical symbols. An expression like x⁻² or √x is an algebraic fraction or radical expression, not a polynomial.