Degree

Algebra-1

1. Fundamental Concepts

The degree of a polynomial refers to the highest degree among all its individual terms (the degree of a single term is the sum of exponents of all variables in it).
- Key Note: Constant terms (terms with no variables, e.g., 5, -2) have a degree of 0, so they do not affect the polynomial’s degree; if a polynomial only has a constant term (e.g., "3"), its degree is 0.

2. Key Concepts

Key Concept	Definition	Example
Linear Polynomial (1st-degree)	Polynomial with a highest degree of 1	"2x + 3" (term degrees: 1, 0 → highest = 1), "a - 5b" (term degrees: 1, 1 → highest = 1)
Quadratic Polynomial (2nd-degree)	Polynomial with a highest degree of 2	"x² - 4x + 1" (term degrees: 2, 1, 0 → highest = 2), "3xy + 2y" (term degrees: 2, 1 → highest = 2)
Cubic Polynomial (3rd-degree)	Polynomial with a highest degree of 3	"x³ - 2x²y + 5" (term degrees: 3, 3, 0 → highest = 3)
Degree of a Simplified Polynomial	The degree is determined only after combining like terms (unsimplified terms do not count)	Unsimplified: "2x² + 3x - x²" → simplified: "x² + 3x" → degree = 2

3. Examples

(1) Easy Difficulty

Question: What is the degree of the polynomial "5x³ - 7x + 2"?
Solution: Calculate each term’s degree: "5x³" (3), "-7x" (1), "2" (0). Highest degree = 3 → Polynomial degree is 3.

(2) Medium Difficulty

Question: Find the degree of the polynomial "4a²b - 3ab² + b³ - 1".
Solution: Term degrees: "4a²b" (2+1=3), "-3ab²" (1+2=3), "b³" (3), "-1" (0). Highest degree = 3 → Polynomial degree is 3.

(3) Difficult Difficulty

Question: Simplify the polynomial "2(x²y - xy) + 3x³y - x²y" and find its degree.
Solution:
Step 1: Simplify: 2x²y - 2xy + 3x³y - x²y = (2x²y - x²y) - 2xy + 3x³y = x²y - 2xy + 3x³y.
Step 2: Calculate term degrees: "x²y" (2+1=3), "-2xy" (2), "3x³y" (3+1=4). Highest degree = 4 → Polynomial degree is 4.

4. Problem-Solving Techniques

Technique 1: Basic Polynomial Degree Calculation
1. List all terms of the polynomial (include signs, ignore coefficients).
2. Calculate the degree of each term (sum variable exponents; no exponent = 1, no variables = 0).
3. Pick the largest degree → that’s the polynomial’s degree.
Technique 2: Degree Calculation After Simplification
1. First simplify the polynomial: remove parentheses (follow sign rules) and combine like terms.
2. Repeat the "list terms → calculate term degrees → find the highest" steps for the simplified polynomial.
- Warning: Do not use unsimplified terms (e.g., "3x² - x²" must be simplified to "2x²" before calculating degree).
Technique 3: Multivariable Polynomial Degree

For polynomials with multiple variables (e.g., a, b; x, y), calculate each term’s degree by summing all variables’ exponents in the term, then take the highest sum as the polynomial’s degree.
Example: Polynomial "2x²y + 5xy² - 3x" → term degrees: 3 (2+1), 3 (1+2), 1 → degree = 3.