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Zero Polynomial — Definition, Formula & Examples

The zero polynomial is the polynomial whose value is 0 for every input — written simply as p(x)=0p(x) = 0. It is the only polynomial that has no nonzero terms.

The zero polynomial is the element of the polynomial ring in which every coefficient equals zero. By convention, its degree is either left undefined or assigned the value −∞-\infty, distinguishing it from all other polynomials (including nonzero constant polynomials, which have degree 0).

Key Formula

p(x)=0p(x) = 0
Where:
  • p(x)p(x) = The zero polynomial, equal to 0 for all values of x

Worked Example

Problem: Determine whether f(x)=3x2−3x2+5x−5xf(x) = 3x^2 - 3x^2 + 5x - 5x is the zero polynomial, and state its degree.
Combine like terms: Group the x2x^2 terms and the xx terms separately.
f(x)=(3x2−3x2)+(5x−5x)=0+0=0f(x) = (3x^2 - 3x^2) + (5x - 5x) = 0 + 0 = 0
Identify the polynomial: Every coefficient has simplified to 0, so this is the zero polynomial.
State the degree: By convention the degree of the zero polynomial is undefined (or −∞-\infty), not 0.
Answer: f(x)f(x) is the zero polynomial, and its degree is undefined.

Why It Matters

The zero polynomial acts as the additive identity in polynomial arithmetic — adding it to any polynomial leaves that polynomial unchanged. Recognizing it prevents errors when computing degrees or counting roots, especially in precalculus and linear algebra.

Common Mistakes

Mistake: Saying the zero polynomial has degree 0, just like other constant polynomials.
Correction: A nonzero constant like p(x)=5p(x) = 5 has degree 0, but the zero polynomial's degree is undefined (or −∞-\infty). This special case matters when applying theorems about polynomial degree.