How Many Zeros in a Quartic Polynomial?
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). For real-coefficient quartics, the real zeros can number 0, 2, or 4. Quartic is the highest degree for which a general algebraic solution formula exists. Related: Polynomial function zeros.
A quartic polynomial has
4
zeros
- Written Form
- f(x) = ax⁴ + bx³ + cx² + dx + e (where a ≠ 0)
- Scientific
- Degree 4
The Real/Complex Breakdown for a Quartic
The direct answer: always exactly 4. For real coefficients, the possible distributions are: See also: Even degree polynomial zeros.
| 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.
Is Quartic Degree 4 or 5?
Quartic means degree 4, from the Latin quartus (fourth). Degree 5 is quintic.
- Quartic
- Degree 4 — "quartus," fourth
- Biquadratic
- A quartic with only even powers (ax4 + bx2 + c), solvable via u = x2
How Do You Find the Zeros of a Quartic Polynomial?
- Rational Root Theorem + synthetic division: Reduce to a cubic or quadratic
- Biquadratic substitution: Substitute u = x2 to get a solvable quadratic in u
- Factoring into two quadratics
- Ferrari's method: The general quartic formula — the highest degree where such a formula exists
For instance,
f(x) = x4 − 5x2 + 4is biquadratic: substituting u = x2 gives (u − 1)(u − 4) = 0, so x = ±1 and x = ±2 — four distinct real zeros.
Zeros in a Quartic Polynomial: FAQ
Can a quartic have exactly 1 or 3 real zeros?
Quick answer
No — since complex roots come in pairs, the real zero count for a quartic is always 0, 2, or 4.
Who solved the general quartic formula?
Quick answer
Lodovico Ferrari, in the 16th century — his method is the highest-degree general algebraic solution that exists.
What is a biquadratic?
A quartic with only even-power terms (ax4 + bx2 + c), solvable by substituting u = x2.
Can a quartic have zero real zeros?
Yes — for example, f(x) = x4 + 1 has all four zeros complex.
Why is quartic the highest degree with a general formula?
The Abel-Ruffini theorem proves no general radical formula exists for degree 5 and above.