Calculus How To

Zero Polynomial: Definition

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What is a Zero Polynomial?

The zero polynomial (also called the zero function) has several different definitions, depending on the author. For example, it is sometimes defined as a polynomial with degree of 0 (where the degree of a polynomial is the largest exponent of any term with a nonzero coefficient). However, it could also mean:

  • A degree of -1.
  • A degree of infinity (∞),
  • An undefined degree (i.e., no degree is assigned to it),
  • Simply the function f(x) = 0.
  • A formal sum with all zero coefficients (by convention, -∞); This comes about because when all of a polynomials coefficients are zero (sometimes called “padding” with zero terms), the polynomial disappears and becomes f = 0,

One reason for these different definitions is a matter of convenience: depending on what mathematical operation you’re performing, you may want to assign the zero polynomial a degree of -1 or -∞, but as neither of these functions are easy to manipulate it’s usually easier just to leave its degree undefined.

A non-zero polynomial has a degree of 1 or more (not including infinity).

A Special Property of the Zero Polynomial

The zero polynomial has a special property: it is the only polynomial that has a function value that becomes close to zero as x gets sufficiently large.

zero polynomial

Function values for the zero polynomial f(x)=x-1 get closer to zero as the x-values get larger.


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[1] University of Utah. Mathematics 1010 online: Polynomials. Retrieved October 24, 2021 from:
[2] Aspnes, J. (2003). Polynomials. Retrieved October 24, 2021 from:
[3] Retrieved October 24, 2021 from:
[4] Bertram. Polynomial Basics. Retrieved October 24, 2021 from:
[5] Rings of Polynomials. Retrieved October 24, 2021 from:

Stephanie Glen. "Zero Polynomial: Definition" From Calculus for the rest of us!

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