How Many Zeros in a Polynomial with Imaginary Zeros?
By Vinitha Published Updated
How many zeros in a polynomial with imaginary zeros? Yes — the zeros of a polynomial can be imaginary (complex) numbers. A zero of a polynomial f(x) is any value of x for which f(x) = 0, and the FTA guarantees all such values exist in the complex number system. Complex zeros take the form a + bi (where b ≠ 0 and i = √−1). For real-coefficient polynomials, complex zeros always appear in conjugate pairs. To see biquadratic substitution with u = x² and the highest degree with a general solution formula, read how many zeros in a quartic polynomial.
A polynomial with imaginary zeros has
Complex pairs
zeros
- Written Form
- Zeros of form a + bi and a - bi
- Scientific
- Conjugate pairs
Can Zeros of a Polynomial Be Imaginary?
Yes — polynomials frequently have imaginary (complex) zeros. The simplest example is f(x) = x2 + 1. Setting x2 + 1 = 0 gives x2 = −1, so x = ±i — purely imaginary zeros with no real part.
| Polynomial | Zeros | Type |
|---|---|---|
| x² + 1 | ±i | Purely imaginary |
| x² + 2x + 5 | −1 ± 2i | Complex (non-real) |
| x² − 4 | ±2 | Real |
| x⁴ + 1 | ±(1±i)/√2 | All complex |
A polynomial of degree 4 can have 0, 2, or 4 complex zeros. More on the Fundamental Theorem of Algebra: how many zeros in a polynomial function.
How Do You Find the Imaginary Zeros of a Polynomial?
For quadratics, the quadratic formula reveals complex zeros when the discriminant b² − 4ac is negative. Taking the square root of a negative number introduces i.
- Factor and reduce
- Find real roots first, then solve the remaining quadratic factor if its discriminant is negative
- Conjugate pairs rule
- If a + bi is a root, a − bi is also a root — use this to construct and divide out the quadratic factor
- Numerical methods
- Newton's method and similar algorithms locate complex roots approximately for high-degree polynomials
How Many Imaginary Zeros Can a Polynomial Have?
- A polynomial of degree n can have at most n complex zeros
- For real-coefficient polynomials, the complex-zero count is always even: 0, 2, 4, … up to n
- An odd-degree polynomial can have at most n − 1 complex zeros, since at least 1 real zero is guaranteed
Graphically, imaginary zeros have no x-intercepts — a graph with all complex zeros never crosses the x-axis.
Watch: How Many Zeros in a Polynomial with Imaginary Zeros?
How many zeros in a polynomial with imaginary zeros? Watch this short video for the zero count, the written form and the notation at a glance.
Imaginary Zeros in a Polynomial: FAQ
Do complex zeros always come in pairs?
Quick answer
Yes, for polynomials with real coefficients — if a + bi is a zero, a − bi must also be one.
Can a linear polynomial have an imaginary zero?
Quick answer
No — a linear polynomial with real coefficients always has exactly one real zero. Questions about the discriminant and quadratic formula are answered in zeros in a quadratic polynomial.
What's the difference between "imaginary" and "complex"?
A complex number has the form a + bi; it's "purely imaginary" only when a = 0.
Can a cubic have all complex zeros?
No — an odd-degree polynomial with real coefficients always has at least one real zero.
Do imaginary zeros show up on a graph?
No — since they're not real numbers, they never appear as x-intercepts.
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.