How Many Zeros in a Polynomial Function?
A polynomial function of degree n has exactly n zeros (roots) in the complex number system, counting multiplicity — this is guaranteed by the Fundamental Theorem of Algebra. These zeros may be real numbers, complex numbers (involving i = √−1), or repeated values. A degree-3 polynomial has 3 zeros; a degree-5 polynomial has 5 zeros; and so on. The number of distinct real zeros can be fewer than n, but the total count of zeros (real and complex, with multiplicity) always equals n exactly. See also: How many zeros in an odd degree polynomial.
A polynomial function has
≤ n
zeros
- Written Form
- f(x) = aₙxⁿ + ... + a₁x + a₀
- Scientific
- Degree n
How Many Zeros Does a Polynomial Function Have?
The number of zeros equals the degree of the polynomial. A degree-n polynomial has exactly n zeros when all complex roots are included and repeated roots are counted according to their multiplicity. The table below shows how degree maps to zero count: See also: Zeros in numbers reference and guide.
| Degree | Type | Zeros (total) | Max real zeros |
|---|---|---|---|
| 0 | Constant | 0 (or infinite if f(x)=0) | 0 |
| 1 | Linear | 1 | 1 |
| 2 | Quadratic | 2 | 2 |
| 3 | Cubic | 3 | 3 |
| 4 | Quartic | 4 | 4 |
| n | Degree-n | n | n |
A root with multiplicity greater than 1 is counted multiple times. For example, f(x) = (x − 3)2 has degree 2 and has the zero x = 3 counted twice (multiplicity 2), so it still has exactly 2 zeros total.
What Is a Zero of a Polynomial Function?
A zero (also called a root) of a polynomial function f(x) is any value of x for which f(x) = 0. Geometrically, zeros are the x-intercepts of the polynomial's graph — the points where the curve crosses or touches the x-axis. A zero with odd multiplicity crosses the x-axis; a zero with even multiplicity touches the axis and bounces back without crossing.
For example, f(x) = x2 − 4 has zeros at x = 2 and x = −2, because f(2) = 0 and f(−2) = 0. Both are real zeros and both are x-intercepts. By contrast, f(x) = x2 + 4 has no real zeros — its two zeros are complex: x = 2i and x = −2i — and its graph never crosses the x-axis.
How Do You Find the Zeros of a Polynomial Function?
The method depends on the degree. For low-degree polynomials:
- Degree 1 (linear): Solve ax + b = 0 directly → x = −b/a
- Degree 2 (quadratic): Factor, complete the square, or use the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
- Degree 3–4: Try rational root theorem, synthetic division, or factoring; cubic and quartic formulas exist but are rarely used
- Degree 5+: No general algebraic formula exists (Abel-Ruffini theorem); numerical methods (Newton's method, graphing) are used
Graphically, zeros are found by reading the x-intercepts. Any value where the graph crosses y = 0 is a real zero of the polynomial.
How Many Polynomials Can Have the Same Zeros?
Infinitely many. If a polynomial has zeros at x = 4 and x = 7, one such polynomial is f(x) = (x − 4)(x − 7) = x2 − 11x + 28. But 2(x − 4)(x − 7), 5(x − 4)(x − 7), and any constant multiple produce a different polynomial with the same zeros. Adding higher-degree factors like (x − 4)(x − 7)(x2 + 1) also gives zeros at 4 and 7 (plus complex zeros). A set of zeros alone does not uniquely define a polynomial — the leading coefficient and any additional roots are also needed.