Polynomial Degree Calculator

What is the degree of a polynomial?

The degree is the highest exponent of the variable after like terms are combined. A polynomial does not have to be written in order, so the safest method is to scan every term, combine matching powers if needed, and then choose the highest remaining power. OpenStax defines the degree and leading coefficient in its lesson on identifying polynomial degree and leading coefficient.

This calculator is built for classroom-style expressions like 4x³ - 2x⁵ + 7x - 9. It identifies the highest power, writes the polynomial in descending order, and names the leading term so the answer is more useful than a single number.

How to find the degree by hand

Terms with negative exponents, variables in denominators, square roots of variables, or fractional powers are not ordinary polynomials. For example, x⁻² + 3 and √x + 4 are not polynomial expressions in the standard algebra sense.

Degree names students should know

Why degree matters

The degree helps you predict graph behavior, possible number of roots, possible turning points, and the type of algebra techniques that may apply. A quadratic can have at most two real roots, a cubic can have at most three, and a fifth-degree polynomial can have at most five. Degree is also the first thing you need when using the leading coefficient test for end behavior.

Common questions

• How do I find the degree of a polynomial?
  Look for the highest exponent of the variable after combining like terms. If the expression is 3x² + 5x⁴ - 7, the highest exponent is 4, so the degree is 4. If the polynomial is not written in descending order, you still choose the highest power, not the first term.
• What is the degree of a constant polynomial?
  A nonzero constant such as 8 has degree 0 because it can be thought of as 8x⁰, and x⁰ equals 1. The zero polynomial is a special case. Many textbooks say its degree is undefined because there is no leading nonzero term.
• Can a polynomial have a negative degree?
  No ordinary polynomial has a negative degree. Polynomial exponents must be nonnegative whole numbers. An expression such as x⁻¹ + 2 is not a polynomial because x⁻¹ means 1/x. If you see negative or fractional powers, you are dealing with a different type of algebraic expression.
• Is the leading coefficient always the first number?
  Only if the polynomial is already written in descending powers. In 5 - 7x + 2x⁴, the leading term is 2x⁴ and the leading coefficient is 2, even though 5 appears first. The leading coefficient belongs to the term with the highest degree.
• Why do I need the degree of a polynomial?
  The degree tells you a lot about the polynomial before you graph or solve it. It helps predict end behavior, maximum possible number of roots, and possible number of turning points. Teachers often ask for degree because it is a foundation for later questions about graphs and factoring.
• What if two highest-degree terms cancel?
  You must combine like terms first. For example, 3x⁴ - 3x⁴ + 2x² + 1 simplifies to 2x² + 1, so the degree is 2, not 4. This is why a calculator should combine like powers instead of only scanning the original expression.