How Many Zeros in a Sextic Polynomial?
By Vinitha Published Updated
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 is not guaranteed to have any real zeros at all. Questions about the rational root theorem and synthetic division are answered in zero count of a polynomial function.
A sextic polynomial has
6
zeros
- Written Form
- f(x) = ax⁶ + bx⁵ + cx⁴ + dx³ + ex² + fx + g (where a ≠ 0)
- Scientific
- Degree 6
The Real/Complex Breakdown for a Sextic
The direct answer: always exactly 6. For real coefficients, the real/complex breakdown must have an even number of complex roots: Questions about positive leading coefficients and minimum values are answered in zeros in an even degree polynomial.
| Real zeros | Complex zeros |
|---|---|
| 6 | 0 |
| 4 | 2 |
| 2 | 4 |
| 0 | 6 |
Unlike odd-degree polynomials, 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 comes from the Latin sextus (sixth); "hexic" is the Greek-derived alternative. A "bicubic" sextic like f(x) = 2x6 − 3x4 + x2 − 5 can be approached with the substitution u = x2 to reduce it to a cubic.
Polynomial Degree Naming Scale
- Degree 4
- Quartic
- Degree 5
- Quintic
- Degree 6
- Sextic (hexic)
- Degree 7
- Septic (heptic)
What Comes After Sextic in the Polynomial Degree Scale?
- Degree 7 — Septic (heptic)
- Degree 8 — Octic
- From degree 5 onward, no general formula using radicals exists (Abel-Ruffini theorem)
The quartic is the highest degree with a complete algebraic solution formula — making the sextic firmly in the territory where numerical methods are typically required.
Watch: How Many Zeros in a Sextic Polynomial?
How many zeros in a sextic polynomial? A sextic polynomial has 6 zeros. Watch this short video for the zero count, the written form and the notation at a glance.
Zeros in a Sextic Polynomial: FAQ
Can a sextic have zero real zeros?
Quick answer
Yes — unlike odd-degree polynomials, a sextic can have all 6 zeros complex, as in f(x) = x6 + 1.
Is "hexic" the same as "sextic"?
Quick answer
Yes — both terms describe a degree-6 polynomial and are used interchangeably in algebra.
Is there a general formula for solving a sextic?
No — the Abel-Ruffini theorem rules out a general radical formula for any degree 5 or higher.
What's a "bicubic" sextic?
A sextic with only even-power terms, solvable via the substitution u = x2 to reduce it to a cubic.
What comes after sextic by degree?
Septic (degree 7), then octic (degree 8).
About the Author
Designer and Developer · Palakkad, Kerala, India
Vinitha holds a bachelor's degree in Textile and Apparel Design from Avinashilingam University, plus a C-Tech certificate in Graphic Design from Image Creative Education. She builds free, no-clutter reference tools like How Many Zeros. Read her full bio.