Polynomial Root Finder
Enter the coefficients of a polynomial from degree 1 to 10. This online polynomial root finder returns every real and non-real complex root, groups repeated roots, shows an approximate factorization, and verifies the answer by substitution.
Written and checked by TheToolNet Editorial Team · Last reviewed: August 21, 2026 · How we check math tools
A number r is a root (or zero) of polynomial P when substituting r for x makes the polynomial equal zero.
Find Polynomial Roots Online
Choose the highest power, then enter every coefficient in descending order. Use 0 for a missing term. For example, x³ − 4x + 1 uses coefficients 1, 0, −4, 1.
How to use the polynomial roots finder
- Select the degree—the largest exponent with a non-zero coefficient.
- Enter coefficients from the highest power down to the constant.
- Enter zero when a power is absent. For 2x⁴ − 7x + 3, enter 2, 0, 0, −7, 3.
- Select Find all roots. Read the roots, multiplicities, approximate factors and substitution check.
This coefficient-based entry avoids ambiguity about parentheses and multiplication signs. Decimal and negative coefficients are accepted.
How the polynomial root finder works
Linear and quadratic equations
A linear equation ax + b = 0 has root −b/a. A quadratic ax² + bx + c = 0 uses x = (−b ± √(b² − 4ac))/(2a). A negative discriminant produces a complex-conjugate pair.
Cubic through degree 10
For higher degrees, the calculator normalizes the leading coefficient and numerically approximates all roots in the complex plane at the same time. It then polishes each answer and evaluates the original polynomial at every root. The displayed residual |P(r)| lets you check how close the approximation comes to zero.
There is no general solution by radicals for every polynomial of degree five or higher. Numerical root-finding is therefore the practical general method, not a shortcut around a missing displayed formula.
Worked polynomial roots example
Consider P(x) = x³ − 6x² + 11x − 6. Testing the integer factors of 6 reveals that P(1) = 0, so (x − 1) is a factor. Dividing leaves x² − 5x + 6, which factors as (x − 2)(x − 3).
The three roots are x = 1, 2 and 3.
Real, complex and repeated polynomial roots
| Root type | Meaning | Example |
|---|---|---|
| Real root | A root on the real number line. | x² − 4 has roots −2 and 2. |
| Non-real complex root | A root with a non-zero imaginary part. Real-coefficient polynomials contain these in conjugate pairs. | x² + 1 has roots i and −i. |
| Repeated root | The same factor occurs more than once; its count is the multiplicity. | (x − 1)³ has root 1 with multiplicity 3. |
The fundamental theorem of algebra says that a degree n polynomial has n complex roots when multiplicities are counted. That is why this tool reports complex roots instead of claiming that a polynomial such as x² + 1 has “no solution.”
Accuracy and limitations
Roots above degree two are displayed as decimal approximations. Repeated roots and roots packed very close together are numerically sensitive: small coefficient changes can move them noticeably. Always check the residual shown with the result before using an approximation in further work.
Frequently asked questions
What is the difference between a root and a zero?
For a polynomial, the terms are normally interchangeable: r is a root of P(x) = 0 and a zero of the function P when P(r) = 0.
Why must missing powers have a zero coefficient?
Coefficient position identifies the power. In x³ − 4x + 1, the zero between 1 and −4 tells the calculator that the x² term is absent.
Can an odd-degree polynomial have no real roots?
No. A real-coefficient polynomial of odd degree has at least one real root. Its remaining non-real roots occur in conjugate pairs.
Why are some roots approximate?
General higher-degree equations require numerical methods. The calculator shows decimal roots and verifies them by substituting them into the original polynomial.
Can I use this instead of a TI-89 polynomial root finder?
Yes for numeric roots of real-coefficient polynomials through degree 10. If you need help with the calculator app itself, see our TI-89 polynomial root finder guide.
Related tools and references
Divide a polynomial before finding its roots with the polynomial long division calculator. Working on matrices? Try the matrix diagonalization calculator or adjoint matrix calculator. For background, see Polynomial Roots at Wolfram MathWorld. TI-89 users can follow our device-specific steps.