How Many Zeros in a Quartic Polynomial?
A quartic polynomial has exactly 4 zeros in the complex number system, counting multiplicity. A quartic has the form f(x) = ax4 + bx3 + cx2 + dx + e (a ≠ 0), and the Fundamental Theorem of Algebra guarantees exactly 4 roots. For real-coefficient quartics, the real zeros can number 0, 2, or 4 — since complex roots come in conjugate pairs, an even-degree polynomial can have all its roots be complex. Quartic is the highest degree for which a general algebraic solution formula exists. See also: Zero count of a linear polynomial.
A quartic polynomial has
4
zeros
- Written Form
- f(x) = ax⁴ + ... (where a ≠ 0)
- Scientific
- Degree 4
How Many Zeros Are in a Quartic Polynomial?
A quartic polynomial always has exactly 4 zeros. For real coefficients, the possible distributions of real vs. complex zeros are:
| Real zeros | Complex zeros | Example type |
|---|---|---|
| 4 distinct real | 0 | Graph crosses x-axis 4 times |
| 2 real + 1 repeated real | 0 | 3 distinct intercepts (one tangent) |
| 2 real + 2 complex | 2 | Graph crosses x-axis twice |
| 0 real + 4 complex | 4 | Graph never crosses x-axis |
A quartic can have 0, 2, or 4 real zeros — never 1 or 3, because complex roots come in pairs for real-coefficient polynomials. Related: Polynomial function zeros.
Is Quartic Degree 4 or 5?
Quartic means degree 4. The naming convention follows Latin/Greek prefixes for degree: linear (1), quadratic (2), cubic (3), quartic (4), quintic (5), sextic (6). "Quartic" comes from the Latin "quartus" (fourth). Degree 5 is quintic, from the Latin "quintus" (fifth). Quartic is also called "biquadratic" when it contains only even powers of x (i.e., ax4 + bx2 + c), which can be solved by substituting u = x2 to reduce it to a quadratic. See also: Odd degree polynomial zeros.
How Do You Find the Zeros of a Quartic Polynomial?
Common strategies for quartic zeros include:
- Rational Root Theorem + synthetic division: Find rational roots by testing factors of the constant term over the leading coefficient, then reduce to a cubic or quadratic.
- Biquadratic substitution: If the quartic has the form ax4 + bx2 + c, substitute u = x2 to get a quadratic in u, solve, then back-substitute.
- Factoring into two quadratics: ax4 + … can often be factored as (ax2 + bx + c)(dx2 + ex + f), then each quadratic solved separately.
- Ferrari's method: The general quartic formula — algebraically complex but guarantees a solution in radicals. This is the highest degree where such a formula exists.
For instance, f(x) = x4 − 5x2 + 4 is biquadratic: substituting u = x2 gives u2 − 5u + 4 = (u − 1)(u − 4) = 0, so u = 1 or u = 4, giving x = ±1 and x = ±2 — four distinct real zeros.