# 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.

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

## References

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[1] University of Utah. Mathematics 1010 online: Polynomials. Retrieved October 24, 2021 from: https://www.math.utah.edu/online/1010/poly/
[2] Aspnes, J. (2003). Polynomials. Retrieved October 24, 2021 from: https://www.cs.yale.edu/homes/aspnes/pinewiki/Polynomials.html
[3] Polynomial.java. Retrieved October 24, 2021 from: https://algs4.cs.princeton.edu/code/edu/princeton/cs/algs4/Polynomial.java.html
[4] Bertram. Polynomial Basics. Retrieved October 24, 2021 from: https://www.math.utah.edu/~bertram/courses/4030/Polys.pdf
[5] Rings of Polynomials. Retrieved October 24, 2021 from: https://www.math.uci.edu/~ndonalds/math120b/2poly.pdf

CITE THIS AS:
Stephanie Glen. "Zero Polynomial: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/zero-polynomial/
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