How Many Zeros in a Constant Polynomial?
A non-zero constant polynomial has zero zeros — none at all. A constant polynomial has the form f(x) = c, where c is any fixed non-zero number. Because f(x) = c for every value of x, there is no value of x that makes f(x) equal to zero. The one exception is the zero polynomial, f(x) = 0, which satisfies f(x) = 0 for every real number x — meaning it has infinitely many zeros, not zero. This special case is treated differently from all other polynomials. Related: Zero count of a polynomial function.
A constant polynomial has
0
zeros
- Written Form
- f(x) = c (where c ≠ 0)
- Scientific
- Degree 0
How Many Zeros Does a Constant Polynomial Have?
A constant polynomial (degree 0) has no zeros — with one critical exception:
- Non-zero constant polynomial (e.g., f(x) = 5, f(x) = −3, f(x) = π): 0 zeros. The output is always a fixed non-zero number, so f(x) = 0 is never satisfied.
- Zero polynomial (f(x) = 0): Infinitely many zeros — every real number is a solution. This polynomial is sometimes considered to have undefined or no degree, distinct from degree-0 polynomials.
| Polynomial | Form | Zeros |
|---|---|---|
| Non-zero constant | f(x) = 5 | 0 (none) |
| Zero polynomial | f(x) = 0 | ∞ (every x) |
| Linear | f(x) = 2x + 1 | 1 |
| Quadratic | f(x) = x² − 4 | 2 |
This also shows why the Fundamental Theorem of Algebra applies to polynomials of degree ≥ 1: a degree-n polynomial has exactly n zeros (counting multiplicity). Degree-0 constant polynomials sit outside that pattern. Related: How many zeros in a sextic polynomial.
Does a Constant Polynomial Have a Zero?
For a non-zero constant polynomial, no. A zero of a function f(x) is a value of x where f(x) = 0. If f(x) = 7 for all x, then f(x) can never equal 0 — there is no zero. Graphically, the graph of f(x) = 7 is a horizontal line at height 7, which never crosses the x-axis. Since zeros are x-intercepts, and there are none, the polynomial has no zeros.
This is consistent with the general rule: a polynomial of degree n has at most n zeros. A degree-0 polynomial has at most 0 zeros — which means exactly 0 zeros for the non-zero case. The zero polynomial (f(x) = 0) is the lone exception because its graph is the x-axis itself.
What Is the Degree of a Constant Polynomial?
The degree of a non-zero constant polynomial is 0. The degree of a polynomial is the highest power of the variable with a non-zero coefficient. In f(x) = 5 = 5x0, the highest (and only) power is x0 = 1, so the degree is 0. The zero polynomial, f(x) = 0, is conventionally assigned no degree (or sometimes degree −∞), since it cannot be written with a leading term.