Zeros of a Polynomial Calculator
Ask three different tools for the zeros of x² − 2 and you may well get three different-looking answers: 1.41421356, ±1.414, or √2. Only the last one is the zero. The others are roundings of it, and a rounding is not a number you can put in an exam answer — which is the first thing a zeros of a polynomial calculator ought to get right.
Zeros come in kinds, and the kind matters as much as the value. This zeros of a polynomial calculator sorts them: rational ones as fractions, irrational ones as exact surds, complex ones as a + bi, and — when there is genuinely no exact form — it says so rather than quietly handing you a decimal.
The Four Kinds of Answer
| Kind | Example | Its zeros |
|---|---|---|
| Rational | x² − 5x + 6 | 2 and 3 |
| Irrational | x² − 2 | ±√2, exactly |
| Complex | x² + 1 | i and −i |
| No exact form | x³ − 4x + 7 | −2.589133 and a complex pair |
How to Find the Zeros of a Polynomial
Every zero comes from a factor, so the work is really factoring work.
Factor as far as it will go. Take out the common factor, then look for the patterns — difference of squares, cubes, a trinomial that splits. The factoring polynomials calculator does exactly this half if you want to see it on its own.
Read the linear factors. Each one gives a zero straight away: 2x − 3 gives x = 3⁄2. These are the rational zeros, and the rational zeros theorem calculator lists the candidates worth testing before you divide anything.
Put each quadratic through the formula. The discriminant decides what you get. Positive and a perfect square gives fractions. Positive and not a perfect square gives surds. Negative gives a complex pair.
Stop when a bracket will not break. That is not failure. x² + x + 1 has no rational factorisation and never will.
Real Zeros, and What the Graph Does
A real zero is a place where the curve meets the x-axis. Complex zeros are not on the graph at all, which is why a parabola can float above the axis and still have two zeros.
The calculator draws the real ones on a number line, blue for the ones you can write as a fraction and green for the irrational ones. Seeing them spaced out is worth more than reading them in a list — you can tell at a glance whether they cluster, straddle the origin, or sit all to one side.
What the graph does at each zero — cross or bounce — depends on how many times that zero repeats, and the algebraic multiplicity calculator is built around that question. This page is about which numbers the zeros are.
How Many Zeros Should There Be?
Exactly as many as the degree, once complex zeros are counted and repeats are counted as often as they occur. A cubic has three. A quartic has four. No exceptions.
This is the fastest check on any answer. Found two zeros of a cubic and stopped? You are not finished — there is a third, even if it is complex or repeated. The calculator states the count under every answer for that reason.
When a Zero Has No Exact Form
x³ − 4x + 7 will not factor over the rationals, and its zeros cannot be written as fractions or surds. This is not a limitation of the calculator. For degree five and above it is a theorem: there is no general formula in radicals, proved by Abel and Ruffini nearly two hundred years ago.
So the honest answer is a number, clearly labelled as such. You get −2.589133 and the complex pair, and the calculator tells you why that is all there is. Search for a zeros calculator and the top result will walk you through pressing buttons on a graphing calculator for the same answer — and it will miss the complex zeros entirely, because they are not on the screen to trace.
How to Use This Zeros of a Polynomial Calculator
Type the polynomial and press =. Terms can be in any order, brackets are expanded first, and fractions and decimals both work. The keypad has x, powers, brackets and a fraction bar; the key icon switches to your own keyboard.
The band groups the answer by kind, with the count underneath. The number line shows where the real zeros fall. Below that, the working: what the factorisation gave, which zeros came from linear factors, which came from the quadratic formula, and the degree check at the end.
Worked Examples
| Polynomial | Zeros | Kind |
|---|---|---|
| x³ − 6x² + 11x − 6 | 1, 2, 3 | all rational |
| x² − 2 | −√2, √2 | irrational, exact |
| 2x² − 4x + 1 | (2 − √2)⁄2, (2 + √2)⁄2 | surd over a denominator |
| x³ − 1 | 1, (−1 ± √3 i)⁄2 | one real, two complex |
| x⁵ − x | −1, 0, 1, i, −i | three real, two complex |
| x³ − 4x + 7 | −2.589133, 1.2946 ± 1.0138i | no exact form |
Frequently Asked Questions
How do you find the zeros of a polynomial?
How many zeros does a polynomial have?
What is the difference between a zero and a root?
Can a polynomial have no real zeros?
Why are some zeros irrational?
Can every zero be written exactly?
Is this zeros of a polynomial calculator free?
The zeros of a polynomial are the whole point of factoring it, and knowing which kind you are looking at is half the answer. Factor first, read the linear factors, put the quadratics through the formula, and count what you have against the degree. Use the calculator above while that sequence is still settling in your head.