How Many Zeros in an Even Degree Polynomial?
By Vinitha Published Updated
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. For more on Ferrari's method, see zeros in a quartic polynomial.
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.
| 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). For more on the real and complex zero breakdown of a degree-6 polynomial, see how many zeros in a sextic polynomial.
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
Watch: How Many Zeros in an Even Degree Polynomial?
How many zeros in an even degree polynomial? Watch this short video for the zero count, the written form and the notation at a glance.
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.
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. Questions about multiplicity and x-intercepts are answered in zero count of a polynomial function.
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.
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.