## 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.

## 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

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