43.6 Polynomial Degree

Polynomial Degree is the value describing the overall complexity of a polynomial, determined by identifying the highest degree among all of its nonzero terms, and this value forms the basis for classifying polynomials into named categories such as constant, linear, quadratic, and cubic.

Polynomial Nonzero-Term Degree Inventory

Procedure

Every nonzero term within a polynomial has its individual degree calculated separately, using the established monomial degree rules, producing a complete list of degree values, one for each term.

Example

For $4x^3+2x^{2}y-5$, the individual term degrees are $3$, $3$, and $0$.

Greatest Term Degree Selection

Procedure

Among the full inventory of individual term degrees, the single largest value is selected, and this largest value becomes the overall degree of the entire polynomial.

Example

Continuing the previous inventory of $3$, $3$, and $0$, the greatest value is $3$, so the polynomial's overall degree is $3$.

Nonzero Constant Polynomial Degree

Description

A polynomial consisting of a single nonzero constant term, with no variables present at all, has an overall degree of zero, consistent with the degree of a nonzero constant monomial.

Example

The polynomial $12$ has an overall degree of $0$.

Zero Polynomial Undefined Degree

Description

The polynomial consisting entirely of the value zero, with no nonzero terms at all, is assigned no defined degree, since there are no term degrees available from which to select a greatest value.

Reasoning

Because the degree-determination process relies on selecting the greatest value from an inventory of existing term degrees, an empty inventory, resulting from every term being zero, leaves no value to select, making the degree of the zero polynomial undefined within this scope.

Zero-Term Exclusion from Degree Determination

Rule

Any term with a coefficient of zero is excluded from the inventory of term degrees before the greatest value is selected, since such a term does not genuinely contribute to the polynomial.

Example

If a polynomial theoretically included a term of $0x^5$, this term's degree of $5$ is excluded from consideration, and the overall degree is instead based only on the remaining nonzero terms.

Polynomial Degree Determination

Summary Procedure

Determining a polynomial's overall degree combines all of the previous steps: every term's individual degree is found, zero-coefficient terms are excluded, and the greatest remaining value is selected as the polynomial's degree, unless every term is zero, in which case the degree is undefined.

Constant Polynomial Degree Classification

Classification Rule

A polynomial with an overall degree of zero is classified as a constant polynomial.

Example

Linear Polynomial Degree Classification

Classification Rule

A polynomial with an overall degree of one is classified as a linear polynomial.

Example

Quadratic Polynomial Degree Classification

Classification Rule

A polynomial with an overall degree of two is classified as a quadratic polynomial.

Example

Cubic Polynomial Degree Classification

Classification Rule

A polynomial with an overall degree of three is classified as a cubic polynomial.

Example

Higher-Degree Polynomial Classification

Classification Rule

A polynomial with an overall degree of four or greater does not receive its own individually established name in elementary usage, and is instead referred to directly by its degree value, such as a degree-four polynomial or a degree-five polynomial.

Example