How Many Zeros in a Sextic Polynomial?
A sextic polynomial has exactly 6 zeros in the complex number system, counting multiplicity. A sextic (also called a hexic) is a polynomial of degree 6, with the form f(x) = ax6 + bx5 + cx4 + dx3 + ex2 + fx + g (a ≠ 0). Because the degree is even, a sextic polynomial with real coefficients can have 0, 2, 4, or 6 real zeros — it is not guaranteed to have any real zeros at all. All 6 complex zeros are guaranteed to exist, but some or all may be complex numbers. Related: Polynomial function zeros.
A sextic polynomial has
6
zeros
- Written Form
- f(x) = ax⁶ + ... (where a ≠ 0)
- Scientific
- Degree 6
How Many Zeros Does a Sextic Polynomial Have?
A sextic polynomial always has exactly 6 zeros in the complex number system. For real coefficients, the real/complex breakdown must have an even number of complex roots (since complex roots come in conjugate pairs): Related: How many zeros in a polynomial with imaginary zeros.
| Real zeros | Complex zeros |
|---|---|
| 6 | 0 |
| 4 | 2 |
| 2 | 4 |
| 0 | 6 |
Unlike odd-degree polynomials (which always have at least one real zero), a sextic can have no real zeros at all. For example, f(x) = x6 + 1 has no real solutions — all 6 roots are complex.
What Is a Sextic Polynomial?
A sextic polynomial is any polynomial of degree exactly 6. The name "sextic" comes from the Latin "sextus" (sixth); "hexic" is the alternative Greek-derived name (from "hex," six). Both terms are used interchangeably in algebra. The simplest sextic is f(x) = x6. A more general example is f(x) = 2x6 − 3x4 + x2 − 5, a "bicubic" sextic that can be approached with the substitution u = x2 to reduce it to a cubic in u.
Degree naming conventions for polynomials follow a standard pattern: linear (1), quadratic (2), cubic (3), quartic (4), quintic (5), sextic (6). Beyond degree 6, names become less standardized — degree 7 is sometimes called "septic" or "heptic." Related: How many zeros in a constant polynomial.
What Comes After Sextic in the Polynomial Degree Scale?
The polynomial naming scale runs:
- Degree 4: Quartic
- Degree 5: Quintic
- Degree 6: Sextic (hexic)
- Degree 7: Septic (heptic)
- Degree 8: Octic
From degree 5 onward, no general formula using radicals can solve the polynomial equation (Abel-Ruffini theorem). The quartic (degree 4) is the highest degree with a complete algebraic solution formula — making the sextic firmly in the territory where numerical methods are typically required for finding zeros.