Degree of a Polynomial
In short: The degree of a polynomial is the largest total exponent appearing in any single term, which also reveals its family and how its graph behaves at the far ends.
Algebra 1AlgebraA-APR.A.1
What is Degree of a Polynomial?
Finding the degree of a polynomial starts with inspecting each term alone and summing the exponents on its variables; the biggest sum is the degree of the full expression.
With one variable, that means the largest exponent: \(8x^5 + 3x^2 – x + 7\) has degree 5. In \(2x^3y^2 – 7xy + 5y^4\) the term sums are 5, 2 and 4, so the answer is again 5. A lone constant like 7 has degree 0. The term of highest degree is the leading term, and its number out front is the leading coefficient.
How to spot it
- A question wants the degree of a polynomial so it can be labeled linear, quadratic, cubic or higher.
- You must write an expression in standard form, from the highest power to the lowest.
- A product of two expressions is formed and the degree of the result is requested.
- Terms contain more than one variable and you must add exponents inside each term.
The method
- Combine like terms first so that no term is hiding.
- For each term, add the exponents of all its variables; a lone number counts as 0.
- Choose the largest of these numbers: that is the degree of a polynomial.
- Reorder terms from highest to lowest degree and read the leading coefficient from the first term.
Degree of a Polynomial: solved example
Problem
Arrange \(6 – 4x^2 + 9x^3 – x\) with its powers descending, then report degree, leading coefficient and type.
Solution
Order from the highest exponent down: \(9x^3 – 4x^2 – x + 6\).
Exponent 3 tops the list, which makes the result a cubic: its degree is 3 and its leading coefficient is 9.
Try it
Question 1
What is the degree of the constant polynomial 7?
Show the answer
Degree 0, since \(7 = 7x^0\).
Question 2
Find the degree of the product \((2x^2)(3x^3)\).
Show the answer
The product is \(6x^5\), so the degree is 5 (exponents add when multiplying).
Degree of a Polynomial vs. Coefficient
The coefficient is a number in front of a variable, while the degree is a count that comes from an exponent. In \(9x^3\) the coefficient is 9 and the degree is 3. People often swap them, so remember: coefficients multiply, exponents give the degree.
Keep going
Lessons
Worksheets and quizzes
- Adding, Subtracting and Multiplying Polynomials – Worksheet
- Adding, Subtracting and Multiplying Polynomials – Quiz
- Structure of Algebraic Expressions – Worksheet
- Structure of Algebraic Expressions – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Algebra