Polynomial Root Calculator
Find all real and complex roots of a polynomial, with exact answers, multiplicities, factored form, a graph, and step-by-step working.
Roots of x³ − 6x² + 11x − 6
All roots exact- Degree
- 3
- Type
- Cubic
- Leading coefficient
- 1
- Constant term
- −6
- Terms
- 4
- Monic
- Yes (leading coefficient 1)
- Real roots
- 3
- Non-real complex roots
- 0
- Distinct roots
- 3
- Repeated roots
- None
Factored form
x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3)
Each root r gives a factor (x − r); a root with multiplicity m gives (x − r)ᵐ.
Graph of y = P(x)
Tap or focus a root marker for details. A ring marks a repeated root.
Roots and verification
| Root | Type | Multiplicity | Verification |
|---|---|---|---|
| 1 | Real, rational | 1 | P(1) = 0 exactly |
| 2 | Real, rational | 1 | P(2) = 0 exactly |
| 3 | Real, rational | 1 | P(3) = 0 exactly |
Exact roots are verified with exact arithmetic (including √ and i). Approximate roots are verified by substituting them into P(x) and checking the result is 0 to within rounding.
Step-by-Step Solution
Step 1 — Set the polynomial equal to 0
x³ − 6x² + 11x − 6 = 0
Step 2 — List possible rational roots (Rational Root Theorem)
Any rational root p/q has p dividing the constant term (−6) and q dividing the leading coefficient (1).
Candidates: ±1, ±2, ±3, ±6
Step 3 — Test the candidates: x = 1 is a root
P(1) = 0 ✓, so (x − 1) is a factor.
Step 4 — Divide by (x − 1) with synthetic division
Quotient: x² − 5x + 6, remainder 0
1 1 −6 11 −6 1 −5 6 1 −5 6 0 Bring down the first coefficient, multiply by 1, add to the next column, and repeat. The last number is the remainder (0 confirms the root); the others are the quotient's coefficients.
Step 5 — Solve the remaining quadratic
x² − 5x + 6 = 0
a = 1, b = −5, c = 6
x = [−b ± √(b² − 4ac)] / 2a
D = b² − 4ac = (−5)² − 4 × 1 × 6 = 1
D > 0: two distinct real roots
x = (5 ± √1) / 2
x₁ = 2, x₂ = 3
Step 6 — Roots
x₁ = 1
x₂ = 2
x₃ = 3
Factored form: (x − 1)(x − 2)(x − 3)
For a single quadratic with the formula worked in detail, use the Quadratic Formula Calculator; for ax + b = c, the Linear Equation Calculator. To find where a polynomial is positive or negative, try the Inequality Calculator, and for arithmetic with the complex roots, the Complex Number Calculator. Several equations at once? Use the System of Equations Calculator.
This polynomial root calculator finds every root, or zero, of a polynomial: the values of x that make P(x) = 0. Type a polynomial such as x³ − 6x² + 11x − 6, or enter its coefficients one by one, and it identifies the degree and type, then finds all roots, real and complex, with their multiplicities.
Roots are exact wherever possible: integers, fractions, square roots such as √2, and complex numbers such as 1 + i√3. The results include the factored form, a check that each root gives P(x) = 0, a graph showing where the curve crosses or touches the x-axis, and a step-by-step solution using the quadratic formula, the Rational Root Theorem, and synthetic division. If a root can only be approximated, the calculator says so and marks it with ≈.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Quadratic: x² − 5x + 6 | x = 2, x = 3 | a = 1, b = −5, c = 6, so D = 25 − 24 = 1 and x = (5 ± 1)/2. Factored: (x − 2)(x − 3). |
| Cubic: x³ − 6x² + 11x − 6 | x = 1, 2, 3 | The Rational Root Theorem gives ±1, ±2, ±3, ±6. P(1) = 0, and synthetic division leaves x² − 5x + 6 = (x − 2)(x − 3). |
| Repeated root: x² − 4x + 4 | x = 2 (multiplicity 2) | D = 16 − 16 = 0, so there is one repeated root: (x − 2)². The graph touches the x-axis at 2 without crossing. |
| Complex roots: x² + 4 | x = ±2i | D = −16 < 0, so x = ±√(−16)/2 = ±2i. There are no real roots, and the parabola never meets the x-axis. |
| Higher degree: x⁴ − 16 | x = −2, 2, −2i, 2i | P(2) = 0 and P(−2) = 0; dividing both out leaves x² + 4, with roots ±2i. Factored: (x + 2)(x − 2)(x² + 4). |
What Are Polynomial Roots?
A root (or zero) of a polynomial is a value of x that makes the polynomial equal to zero. For P(x) = x² − 5x + 6, P(2) = 4 − 10 + 6 = 0, so x = 2 is a root, and so is x = 3.
On a graph, the real roots are where the curve meets the x-axis. Some polynomials, such as x² + 4, never meet the x-axis: they have no real roots, but they still have complex roots (here 2i and −2i). A polynomial of degree n always has exactly n roots when complex and repeated roots are counted.
How to Find Polynomial Roots
Factoring works when the polynomial splits into simple factors: x² − 5x + 6 = (x − 2)(x − 3), so the roots are 2 and 3. The quadratic formula, x = [−b ± √(b² − 4ac)] / 2a, solves any quadratic, including ones that do not factor nicely.
For cubics and higher, test the candidates from the Rational Root Theorem, and each time one works, divide it out with synthetic division to get a smaller polynomial. Repeat until a quadratic remains. Not every polynomial has rational roots: x³ − 2x − 5 has none, and its roots can only be found with numerical methods, which give very accurate approximations rather than exact values.
The Rational Root Theorem
If a polynomial has integer coefficients, every rational root p/q (in lowest terms) has p dividing the constant term and q dividing the leading coefficient. For 2x² − 5x + 2, p divides 2 and q divides 2, so the only candidates are ±1, ±2, and ±1/2. Testing them shows P(2) = 0 and P(1/2) = 0, so the roots are 2 and 1/2. The theorem gives a short list to test, but it cannot find irrational or complex roots.
Synthetic Division
Once you know a root r, synthetic division divides the polynomial by (x − r) using only its coefficients. For x³ − 6x² + 11x − 6 and r = 1, write the coefficients 1, −6, 11, −6. Bring down the 1, multiply by 1 and add to −6 to get −5, multiply by 1 and add to 11 to get 6, then multiply by 1 and add to −6 to get 0. The remainder 0 confirms the root, and 1, −5, 6 are the coefficients of the quotient x² − 5x + 6.
Multiplicity and Complex Roots
A root that appears more than once is a repeated root. In x² − 4x + 4 = (x − 2)², the root 2 has multiplicity 2. At a root of even multiplicity the graph touches the x-axis and turns back; at a root of odd multiplicity it crosses.
Complex roots have the form a + bi, where i² = −1. When the coefficients are real, complex roots come in conjugate pairs, a + bi and a − bi, which is why a cubic always has at least one real root: its three roots cannot all be paired up.
How to Use the Polynomial Root Calculator
- Type the polynomial, such as x³ − 6x² + 11x − 6, using ^ or superscripts for powers (x^3 or x³). You can also type an equation such as x² = 4, or switch to Coefficients to enter a₃, a₂, a₁, a₀ one by one.
- Press Solve, or tap an example. The calculator identifies the degree, leading coefficient, and type, then finds every root, real and complex.
- Read the roots: exact roots are shown as integers, fractions, square roots, or a + bi, and numerical approximations are marked with ≈. Repeated roots show their multiplicity.
- Check the factored form, the verification of each root (P(r) = 0), and the graph, where each real root is an x-intercept you can tap for details.
- Follow the step-by-step solution: the quadratic formula, or the Rational Root Theorem with synthetic division for higher degrees, and a clear note whenever numerical root finding was needed.
Frequently Asked Questions
What is a root of a polynomial?
A value of x that makes the polynomial equal zero. For P(x) = x² − 5x + 6, x = 2 is a root because P(2) = 0.
What is the difference between a root and a zero?
They mean the same thing. "Root" usually refers to a solution of the equation P(x) = 0, and "zero" to an input where the function P(x) equals 0. Real roots are also the x-intercepts of the graph.
How do you find the roots of a polynomial?
Solve linear polynomials directly and quadratics with the quadratic formula. For higher degrees, test rational candidates from the Rational Root Theorem, divide out each root with synthetic division, and solve the quadratic that remains, or use a numerical method if there are no rational roots.
How many roots can a polynomial have?
A polynomial of degree n has exactly n roots, counting complex roots and repeated roots. It can have at most n distinct real roots.
Can a polynomial have complex roots?
Yes. x² + 4 has the complex roots 2i and −2i. For polynomials with real coefficients, complex roots always come in conjugate pairs a ± bi.
What is a repeated root?
A root that occurs more than once, so its factor appears more than once. x² − 4x + 4 = (x − 2)² has the repeated root x = 2.
What is root multiplicity?
The number of times a root is repeated: the power of its factor. In (x − 2)²(x + 1), x = 2 has multiplicity 2 and x = −1 has multiplicity 1. Even multiplicity means the graph touches the x-axis without crossing it.
How does the quadratic formula find roots?
For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / 2a. The discriminant b² − 4ac tells you the type: positive gives two real roots, zero gives one repeated root, and negative gives two complex conjugate roots.
Why does a cubic polynomial always have at least one real root?
A cubic goes to +∞ at one end and −∞ at the other, so its graph must cross the x-axis at least once. Equivalently, complex roots come in pairs, so one of the three roots must be real.
What does it mean when a polynomial has no real roots?
Its graph never meets the x-axis, as with x² + 4. It still has roots, but they are complex numbers. Only a nonzero constant polynomial has no roots at all.
How do roots relate to factors?
x = r is a root exactly when (x − r) is a factor. So the roots 1, 2 and 3 of x³ − 6x² + 11x − 6 give the factorization (x − 1)(x − 2)(x − 3).
What is the difference between exact and approximate roots?
Exact roots are written precisely, such as 3, 1/2, √2, or 1 + i√3. Approximate roots, such as x ≈ 2.094551 for x³ − 2x − 5, are computed numerically when no exact form can be found; this calculator marks them with ≈.
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Last updated: September 27, 2026.