What are the zeros of a polynomial?
A zero is an x value that makes the polynomial equal 0. Zeros, roots, and solutions all mean the same number here, so a roots calculator and a zeros calculator do the same job.
On a graph, each real zero is a point where the curve crosses or touches the x-axis. The polynomial x³ − 6x² + 11x − 6 crosses at x = 1, x = 2, and x = 3, so those are its zeros.
How does the rational root theorem help?
The rational root theorem gives you a short list of every fraction that could be a root. Any rational root p/q must have p dividing the constant term and q dividing the leading coefficient.
For x³ − 6x² + 11x − 6, the constant is −6 and the leading coefficient is 1. The candidates are ±1, ±2, ±3, and ±6. A rational root theorem calculator builds this list for you, but it is quick to do by hand.
When the leading coefficient is not 1, fractions join the list. For 2x³ + 3x² − 8x + 3, the candidates are ±1, ±3, ±1/2, and ±3/2. Its zeros turn out to be −3, 1/2, and 1.
Worked example: zeros of x³ − 6x² + 11x − 6
Test the candidates, divide out the first root you find, and factor the quadratic that is left. This is the same path the calculator shows in its steps.
Dividing out a root lowers the degree by one. This is called deflating the polynomial. Once you are down to a quadratic, you can factor it or use the quadratic formula, so you rarely need to test every candidate.
What is multiplicity?
Multiplicity is the number of times a zero repeats. If (x − 1) appears twice as a factor, then 1 is a zero of multiplicity 2.
Take x³ − 3x + 2. It factors as (x − 1)²(x + 2), so its zeros are 1 and −2, with 1 counted twice. The calculator lists each zero once in the answer, x = −2 or x = 1. You can see the repeat in the steps, where it divides by (x − 1) two times.
Multiplicity changes the graph. At a zero of odd multiplicity the curve crosses the x-axis. At a zero of even multiplicity, like x = 1 here, the curve touches the axis and turns back.
What about irrational roots and approximate answers?
If the last quadratic or simple power has no rational roots, the calculator gives an exact radical when one exists and a decimal when it does not. Exact forms come first.
For x² − 5, the zeros are x = −√5 and x = √5. For x³ − 2, the only real zero is x = ∛2. Switch on complex numbers and it also lists the two complex roots of x³ − 2, shown as decimals: about −0.629961 ± 1.091124i.
Some polynomials have no rational roots and no simple radical form. The cubic x³ − 3x + 1 has three real zeros, approximately −1.879385, 0.347296, and 1.532089. The calculator marks these as approximate so you know they are rounded.
| Polynomial | Zeros | Type |
|---|---|---|
| x³ − 6x² + 11x − 6 | 1, 2, 3 | Exact, rational |
| x² − 5 | −√5, √5 | Exact, irrational |
| x³ − 2 | ∛2 | Exact, irrational |
| x³ − x − 1 | about 1.324718 | Approximate |
Can the zeros calculator solve cubic equations?
Yes. A cubic equation is a degree 3 polynomial set equal to zero, so finding its zeros solves it. The calculator handles polynomials up to degree 6, which covers cubics, quartics, and the polynomial roots problems most algebra and precalculus classes assign.
If your equation is not already in the form polynomial = 0, the solver page will move everything to one side first. The zeros page expects just the polynomial, or f(x) = followed by the polynomial.
What mistakes do students make when finding zeros?
Forgetting the negative candidates is common. The rational root theorem gives ± for every value, and −2 is just as likely as 2. Another slip is leaving out a zero coefficient in synthetic division. For x³ − 3x + 2, the row must be 1, 0, −3, 2.
Students also stop after finding one root. A cubic can have up to three real zeros, so always divide out the root and solve what is left.