About Polynomial Roots Calculator
This calculator finds all roots of a polynomial equation of degree four, five or more. Type x^4 - 5x^2 + 4 = 0 and it returns x = 1, −1, 2 and −2 by repeatedly dividing out rational roots. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
A polynomial of degree n has exactly n roots counting repeats and complex ones. The solver tests rational roots, divides each one out to lower the degree, and stops when a quadratic or a polynomial without rational roots is left. In the quartic above it finds 1, then −1, and the leftover x² − 4 gives ±2.
When there is no rational root, as in x^4 + 1 = 0 or x^5 - x - 1 = 0, the roots come from Durand-Kerner iteration and are shown as decimals. x^4 + 1 = 0 gives four complex roots ±0.7071067812 ± 0.7071067812i, and x^5 - x - 1 = 0 has one real root, about 1.167303978, and four complex ones. Degree five and above has no general formula in radicals, so numbers are the practical answer.
How to use Polynomial Roots Calculator
- Pick a type: Linear, Quadratic, Cubic, Polynomial roots, 2 equations, 3 equations or Discriminant.
- For a single equation, type it in the box using ^ for powers, such as x^3 - 6x^2 + 11x - 6 = 0. The type is detected from what you enter.
- For a system, fill the grid with the coefficients and the constant of each equation, in the order x, y, z, constant.
- Read the roots or the solution at the top of the result.
- Open the steps to follow the working, and look at the graph to see where the curve meets the axis.
Features
- Linear, quadratic, cubic and higher-degree polynomial equations in one variable, with the degree detected automatically.
- Quadratic roots with the discriminant shown and explained, including complex roots in the form a + bi.
- Exact rational roots shown as integers or fractions, such as x = 1/2, and exact radical and complex roots such as x = 1 ± √2 or x = −1 ± 2i.
- Cubic and higher equations solved by the rational root theorem, reducing the degree step by step, with a numeric fallback when no rational root exists.
- Cubic discriminant reported, so you know whether to expect three real roots or one real and two complex.
- Systems of 2 and 3 linear equations solved by Gauss-Jordan elimination, with the augmented matrix and each row operation typeset.
- Detects systems with no solution, and gives a parametric solution such as x = 2 − t, y = t for systems with infinitely many.
- Brackets, fractions and rational equations through the expression engine; values that make a denominator zero are excluded.
- A graph of the left side minus the right side with the real roots marked, which you can zoom, pan and export.
- A standalone discriminant mode for b² − 4ac.
Roots of a degree-n polynomial
Each root r gives a factor, and the factors multiply back to the polynomial.
aₙxⁿ + … + a₀ = aₙ(x − r₁)(x − r₂)…(x − rₙ)Worked examples
x^4 - 5x^2 + 4 = 0 gives x = 1, −1, 2, −2
Divide out x = 1 to get x³ + x² − 4x − 4, then x = −1 to get x² − 4 = 0, so x = ±2.
x^5 - x - 1 = 0 gives One real root x ≈ 1.167303978 and four complex roots
No rational root exists, so the roots are found numerically.
Privacy: how it works
- Equations are solved by code running in your browser tab. They are not sent to Ulkaa or anyone else, and there is nothing to sign in to.
Tips
- Brackets, fractions and rational equations are accepted, so (x + 1)(x + 2) = 0, x/2 + 1 = 4 and (x + 1)/(x - 2) = 3 can be typed as they are.
- Move nothing yourself: write the equation as you have it, with terms on both sides if needed, and the solver collects them.
- Use a decimal or a fraction for coefficients, such as 0.5x + 2 = 5 or 1/2x + 2 = 5.
- In the system grid, leave a cell blank for a zero coefficient.
- For one variable, the equation can use x or any single letter, but only one letter per equation.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your equations are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your equations are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How many roots does a polynomial have?
A polynomial of degree n has n roots when you count complex roots and repeated roots. Only some of them may be real.
Why are some roots only approximate?
Polynomials of degree five and higher have no general closed formula, and many lower-degree ones have irrational roots. Quadratic irrational roots are shown exactly as square roots; the rest are computed to about ten decimal places.
More Equation Solver pages
- Equation SolverSolve linear, quadratic, cubic and polynomial equations and systems of equations, with the working and a graph.
- Quadratic Equation SolverType a quadratic and get its roots, the discriminant and the working.
- Quadratic Formula CalculatorSee the quadratic formula applied to your numbers, line by line.
- Linear Equation SolverSolve for x in an equation with x on one or both sides.
- System of Equations CalculatorEnter the coefficients of two equations and get x and y with each row operation.
- 3 Variable System of Equations SolverSolve three equations in x, y and z and see every row operation.
- Simultaneous Equations SolverSolve two equations at once and check the answer against both.
- Cubic Equation SolverFind all three roots of a cubic, real and complex.
- Discriminant CalculatorEnter a, b and c to get b² − 4ac and the type of roots.