How Many Zeros in an Even Degree Polynomial?
How many zeros in an 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. The number of real zeros in an even-degree polynomial is always even, including zero. Related: Zero count of a polynomial function.
An 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), and so on. See also: How many zeros in a quartic polynomial.
| 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).
How Many Zeros Does an Even-Degree Polynomial Have?
An even-degree polynomial of degree n always has exactly n zeros in total. This is because complex zeros come in conjugate pairs, so removing a complex pair always reduces the real zero count by 2 at a time.
The most extreme even case — where all zeros are complex — occurs with f(x) = x4 + 2x2 + 1 = (x2 + 1)2, which has zeros x = ±i (each with multiplicity 2) and no real zeros at all.
Does an Even-Degree Polynomial Have to Cross the X-Axis?
- Positive leading coefficient
- Both ends of the graph point upward ("U-shape")
- Negative leading coefficient
- Both ends point downward ("hill-shape")
If the minimum value of f(x) is positive, the graph never dips below the x-axis and has no real zeros.
f(x) = x2 + 5— minimum at f(0) = 5, never crosses the x-axisf(x) = x2 − 5— minimum at f(0) = −5, crosses twice at x = ±√5
Zeros in an Even-Degree Polynomial: FAQ
Can an even-degree polynomial have zero real zeros?
Quick answer
Yes — unlike odd-degree polynomials, an even-degree polynomial can have all its roots complex. See also: Zero count of a quintic polynomial.
Is the real zero count always even?
Quick answer
Yes — 0, 2, 4, and so on, up to the degree, since complex roots come in pairs.
What shape does an even-degree polynomial's graph have?
A "U-shape" (both ends up) for a positive leading coefficient, or a "hill-shape" for a negative one.
Can two even-degree polynomials of the same degree differ in real zero count?
Yes — the degree only fixes the total zero count; the specific coefficients determine how many are real.
How is this different from an odd-degree polynomial?
An odd-degree polynomial always has at least one real zero; an even-degree polynomial can have none.