Zeros of a Function Calculator
Find the zeros, also called roots, of a polynomial up to degree 4, including complex zeros.
Real zeros
x = 1, x = 2, x = 3
- Complex zeros
- —
- Function
- f(x) = x³ − 6x² + 11x − 6
- Degree
- 3
The function has 3 real zeros and 0 complex zeros.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| Check of real zero | f(1) is 0 | 0 |
| Check of real zero | f(2) is 0 | 0 |
| Check of real zero | f(3) is 0 | 0 |
How it works
Type a polynomial in one variable with a degree from 1 to 4.
Worked example
f(x) = x³ − 6x² + 11x − 6:
- It factors as (x − 1)(x − 2)(x − 3)
- The real zeros are x = 1, x = 2 and x = 3
- Check: f(2) = 8 − 24 + 22 − 6 = 0
Sources
Frequently asked questions
What are the zeros of a function?
The zeros, or roots, of a function are the values of x that make it equal to 0. On a graph, the real zeros are where the curve crosses the x-axis. The polynomial x² − 5x + 6 has zeros at x = 2 and x = 3.
How does the calculator find them?
It uses the Durand-Kerner method, which refines guesses for all the roots together until they stop changing. It finds every root of a polynomial, real and complex, up to degree 4.
What are complex zeros?
Some polynomials have no real zero for part of their degree. x² + 1 has no real zero, but it has the complex zeros x = i and x = −i. Complex zeros of real polynomials come in matching pairs.
How many zeros can a polynomial have?
A polynomial of degree n has n zeros counting repeats and complex ones. A repeated zero, such as x = 1 in x² − 2x + 1, is counted twice.
Why are some zeros shown as decimals?
Many zeros are irrational. The zeros of x² − 2 are about ±1.414214, the square root of 2. They are shown to six decimal places.