Degree of a Polynomial
Worked Example
Problem: Find the degree of the polynomial 7x4−3x2+5x−9.
Step 1: Identify each term and its degree. The degree of a term is the exponent on its variable (constants have degree 0).
7x4(degree 4),−3x2(degree 2),5x(degree 1),−9(degree 0)
Step 2: Pick the highest degree among all the terms.
max(4,2,1,0)=4
Answer: The degree of 7x4−3x2+5x−9 is 4.
Why It Matters
The degree of a polynomial controls its end behavior on a graph — for instance, an even-degree polynomial's ends point the same direction, while an odd-degree polynomial's ends point opposite directions. It also tells you the maximum number of roots the polynomial can have and the maximum number of turning points (degree minus one). Knowing the degree is essential when classifying polynomials (linear, quadratic, cubic, etc.).
Common Mistakes
Mistake: Ignoring multi-variable terms. In a term like 6x2y3, students often say the degree is 3 (the largest single exponent) instead of adding the exponents.
Correction: The degree of a single term is the sum of all its variable exponents. So 6x2y3 has degree 2+3=5. Then compare across all terms to find the polynomial's degree.
Related Terms
- Degree of a Term — Exponent sum for a single term
- Polynomial — The expression whose degree is measured
- Leading Coefficient — Coefficient of the highest-degree term
- Monomial — A polynomial with exactly one term