How Many Zeros in a Even Degree Polynomial?
An even-degree polynomial of degree n has exactly n zeros in the complex number system, counting multiplicity. Unlike odd-degree polynomials, an even-degree polynomial is not guaranteed to have any real zeros — all n roots can be complex. A polynomial of degree 2 can have 0 or 2 real zeros; degree 4 can have 0, 2, or 4; degree 6 can have 0, 2, 4, or 6. The number of real zeros in an even-degree polynomial is always even (including zero). See also: Zeros in a quadratic polynomial.
A even degree polynomial has
0 to n
zeros
- Written Form
- Polynomials of degree 2, 4, 6, 8...
- Scientific
- Even n
What Is an Even-Degree Polynomial?
An even-degree polynomial is any polynomial whose highest-power term has an even exponent: degree 2 (quadratic), degree 4 (quartic), degree 6 (sextic), degree 8 (octic), and so on. The degree determines how many total zeros the polynomial has, and the parity (even or odd) determines whether real zeros are guaranteed.
| Degree | Name | Total zeros | Possible real zeros |
|---|---|---|---|
| 2 | Quadratic | 2 | 0 or 2 |
| 4 | Quartic | 4 | 0, 2, or 4 |
| 6 | Sextic | 6 | 0, 2, 4, or 6 |
| n (even) | — | n | 0, 2, 4, … n |
For example, f(x) = x2 + 1 has degree 2 but no real zeros — both roots are complex (x = ±i). Meanwhile, f(x) = x2 − 1 has two real zeros (x = ±1). The degree alone doesn't determine real vs. complex; the specific coefficients do. Related: Polynomial function zeros.
How Many Zeros Does an Even-Degree Polynomial Have?
An even-degree polynomial of degree n always has exactly n zeros in total (complex + real, counting multiplicity). The number of real zeros is always even — 0, 2, 4, … up to n. This is because complex zeros of real-coefficient polynomials come in conjugate pairs, so removing a complex pair always reduces the real zero count by 2 at a time.
This contrasts with odd-degree polynomials, which must have at least 1 real zero. The most extreme even case — where all zeros are complex — occurs, for example, with f(x) = x4 + 2x2 + 1 = (x2 + 1)2, which has zeros x = ±i (each with multiplicity 2) and no real zeros at all. Learn more about zero count of a cubic polynomial.
Does an Even-Degree Polynomial Have to Cross the X-Axis?
No — an even-degree polynomial with a positive leading coefficient has both ends of its graph pointing upward. If the minimum value of f(x) is positive, the graph never dips below the x-axis and has no real zeros. For example, f(x) = x2 + 5 has a minimum at f(0) = 5 > 0, so it never crosses the x-axis. In contrast, f(x) = x2 − 5 has a minimum at f(0) = −5 < 0, so it does cross the axis — twice (at x = ±√5).
This "U-shape" behavior (or "hill-shape" for negative leading coefficients) means an even-degree polynomial may or may not have real zeros, depending purely on whether its extreme value (minimum or maximum) crosses or touches the x-axis.