# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/polynomial/polynomial-degree
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Output docs content for large language models.

---

export const metadata = {
  title: "Polynomial Degree",
  description: "Master finding polynomial degrees by identifying the highest power term. Learn monomial degrees, multi-variable cases, and step-by-step examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/04/2025",
  subject: "Polynomial",
};

## Understanding the Degree of a Monomial

Each monomial within a polynomial has a characteristic called **degree**. This degree is determined by the powers (exponents) of its variables.

### Degree of a Single-Variable Monomial

If a monomial has only one variable, like <InlineMath math="ax^n" />, its degree is the exponent of that variable, which is <InlineMath math="n" />.

**Examples:**

- The monomial <InlineMath math="4x^5" /> has a degree of 5.
- The monomial <InlineMath math="0.12x" /> (or <InlineMath math="0.12x^1" />) has a degree of 1.

### Degree of a Multi-Variable Monomial

If a monomial has more than one variable, its degree is the **sum** of all the variable exponents.

**Examples:**

- The monomial <InlineMath math="\frac{3}{4}x^2y^7" /> has a degree of <InlineMath math="2 + 7 = 9" />.
- The monomial <InlineMath math="2.17x^3yz^3" /> (remember <InlineMath math="y = y^1" />) has a degree of <InlineMath math="3 + 1 + 3 = 7" />.

### Degree of a Constant

What about a constant (a number without variables), like 5? A non-zero constant is considered to have a **degree of 0**, because we can write it as <InlineMath math="5x^0" /> (since <InlineMath math="x^0 = 1" />).

Here is a summary of monomial degree examples in a table:

| Monomial                                | Degree | Explanation                                                       |
| :-------------------------------------- | :----- | :---------------------------------------------------------------- |
| <InlineMath math="4x^5" />              | 5      | The exponent of <InlineMath math="x" /> is 5.                     |
| <InlineMath math="\frac{3}{4}x^2y^7" /> | 9      | Sum of exponents <InlineMath math="2+7=9" />.                     |
| <InlineMath math="0.12x" />             | 1      | The exponent of <InlineMath math="x" /> is 1.                     |
| <InlineMath math="2.17x^3yz^3" />       | 7      | Sum of exponents <InlineMath math="3+1+3=7" />.                   |
| <InlineMath math="10" />                | 0      | Non-zero constant. Can be written as <InlineMath math="10x^0" />. |

### Definition of Monomial Degree

The degree of a monomial (with a non-zero coefficient) is the sum of the exponents of all its variables. For a monomial <InlineMath math="ax^n" />, its degree is <InlineMath math="n" />.

## Determining the Degree of a Polynomial

Once we know how to determine the degree of each monomial (term), finding the degree of a polynomial becomes easier.

**The degree of a polynomial** is the **highest degree** among all the terms (monomials) that make up the polynomial.

**Steps to determine the degree of a polynomial:**

1.  Identify all the terms (monomials) in the polynomial.
2.  Determine the degree of each term.
3.  Choose the highest degree among all the terms. That is the degree of the polynomial.

**Example 1:**

Determine the degree of the following polynomial:

<BlockMath math="8x^3 - 36x^2 + 54x - 27" />

- The term <InlineMath math="8x^3" /> has degree 3.
- The term <InlineMath math="-36x^2" /> has degree 2.
- The term <InlineMath math="54x" /> (or <InlineMath math="54x^1" />) has degree 1.
- The term <InlineMath math="-27" /> (constant) has degree 0.

The highest degree among the terms is 3. Therefore, the degree of this polynomial is **3**.

**Example 2:**

Determine the degree of the following polynomial:

<BlockMath math="5x^4y^2 + xy^2 - 2x^5y^6" />

- The term <InlineMath math="5x^4y^2" /> has degree <InlineMath math="4 + 2 = 6" />.
- The term <InlineMath math="xy^2" /> (or <InlineMath math="x^1y^2" />) has degree <InlineMath math="1 + 2 = 3" />.
- The term <InlineMath math="-2x^5y^6" /> has degree <InlineMath math="5 + 6 = 11" />.

The highest degree among the terms is 11. Therefore, the degree of this polynomial is **11**.

**Example 3:**

Determine the degree of the following polynomial:

<BlockMath math="0.13x^3 + 1.56x^2 - 2.24x + 1.72" />

- The term <InlineMath math="0.13x^3" /> has degree 3.
- The term <InlineMath math="1.56x^2" /> has degree 2.
- The term <InlineMath math="-2.24x" /> has degree 1.
- The term <InlineMath math="1.72" /> has degree 0.

The highest degree is 3. Therefore, the degree of this polynomial is **3**.

### Definition of Polynomial Degree

The degree of a polynomial is the highest degree of its terms.

### What About the Degree of Zero?

Is the degree of 0 equal to 0, since 0 can be written as <InlineMath math="0x^0" />?

Generally in mathematics:

- **Non-zero** constants (like 5, -27, 1.72) have a degree of **0**.
- The **zero polynomial** (the number 0 itself) is often considered to **have no degree** or sometimes is said to have a degree of **negative infinity** (<InlineMath math="-\infty" />). The reason is a bit complex, but essentially it helps keep properties of degrees (like the degree of the product of two polynomials) consistent.

However, for the high school level, understanding that non-zero constants have degree 0 and the degree of a polynomial is the highest degree of its terms is sufficient.
