How-to · Updated October 2, 2026
How do you write a polynomial with given zeros?
Direct answer
Multiply (x − zero) once for each zero, then by any leading coefficient. Zeros −2 and 5 give (x + 2)(x − 5) = x² − 3x − 10. A complex zero needs its conjugate: 2 + 3i produces x² − 4x + 13.
From zeros to a polynomial
| Zeros | Factored | Expanded |
|---|---|---|
| −2 and 5 | (x + 2)(x − 5) | x² − 3x − 10 |
| 3 and 6 | (x − 3)(x − 6) | x² − 9x + 18 |
| −2, 1 and 3, leading coefficient 2 | 2(x + 2)(x − 1)(x − 3) | 2x³ − 4x² − 10x + 12 |
| 1/4 and −1 | (x − 1/4)(x + 1) | 4x² + 3x − 1 |
| 2 + 3i (conjugate added) | (x − (2 + 3i))(x − (2 − 3i)) | x² − 4x + 13 |
Zeros 1/4 and −1 expand to x² + (3/4)x − 1/4. The integer-coefficient form with the same zeros is 4x² + 3x − 1. The row for 2 + 3i includes the conjugate 2 − 3i.
Leading coefficient
Zeros −2, 1 and 3 with leading coefficient 2 give 2(x + 2)(x − 1)(x − 3) = 2x³ − 4x² − 10x + 12. Without that 2, the monic polynomial (x + 2)(x − 1)(x − 3) has the same zeros and half the coefficients. Infinitely many polynomials share a zero list; the leading coefficient is what picks one of them.
Limits
- Each listed zero is simple unless you repeat it. A repeated zero needs the factor squared, or cubed, to match the multiplicity.
- Real coefficients force complex zeros to arrive in conjugate pairs. The calculator adds a missing conjugate.
- These expansions use the same builder as the zeros calculator.
Use the calculator
The other direction: how to find the zeros of a polynomial. Build one in the zeros calculator.