How Many Zeros in a Constant Polynomial?
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. The one exception is the zero polynomial, f(x) = 0, which has infinitely many zeros — every real number is a solution. Learn more about cubic polynomial zeros.
A constant polynomial has
0
zeros
- Written Form
- f(x) = c (where c ≠ 0)
- Scientific
- Degree 0
Why a Constant Polynomial Has No Zeros
A constant polynomial (degree 0) has no zeros — with one critical exception:
- Non-zero constant polynomial (e.g., f(x) = 5, f(x) = −3): 0 zeros. The output is always fixed and non-zero.
- Zero polynomial (f(x) = 0): infinitely many zeros — every real number is a solution.
| 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 shows why the FTA applies to polynomials of degree ≥ 1: a degree-n polynomial has exactly n zeros. Degree-0 constant polynomials sit outside that pattern. See also: Zero count of a quadratic polynomial.
Does a Constant Polynomial Have a Zero?
For a non-zero constant polynomial, no. If f(x) = 7 for all x, then f(x) can never equal 0. Graphically, the graph of f(x) = 7 is a horizontal line at height 7, which never crosses the x-axis.
What Is the Degree of a Constant Polynomial?
- Non-zero constant, e.g. f(x) = 5
- Degree 0 — the highest non-zero power is x0 = 1
- Zero polynomial, f(x) = 0
- Conventionally has no degree (or sometimes degree −∞)
Zeros in a Constant Polynomial: FAQ
Can a constant polynomial ever have a zero?
Quick answer
Only the zero polynomial (f(x) = 0) does — and it has infinitely many, not just one.
Why doesn't the zero-equals-degree rule apply here?
Quick answer
The Fundamental Theorem of Algebra applies to polynomials of degree ≥ 1; degree 0 is a special case outside that pattern. Related: Zeros in a polynomial function.
Is f(x) = 0 really a constant polynomial?
It's the "zero polynomial" — technically distinct from ordinary constant polynomials, since it's conventionally assigned no degree.
Does a constant polynomial's graph ever cross the x-axis?
Only if the constant is 0 — otherwise the graph is a horizontal line that never touches the x-axis.
How many zeros does f(x) = π have?
Zero — π is a fixed non-zero number, so f(x) = π is never equal to 0.