Roots of a Polynomial Calculator
The roots of a polynomial are the values of the variable that make the polynomial equal zero, the points where its graph crosses or touches the horizontal axis. Finding them is a basic task across algebra, calculus and engineering. This calculator accepts a polynomial of up to the fourth degree, entered as the coefficients from the highest power down to the constant term, and returns its real roots. It works by evaluating the polynomial across a wide range, detecting every interval where the value changes sign (a sign change must contain a real root because the curve has to pass through zero), and then refining each interval with the bisection method until the root is pinned down to high precision. Leading coefficients of zero are ignored, so you can solve a cubic or quadratic by leaving the higher coefficients blank or zero. Complex roots are not displayed because this tool reports the real solutions you can read off the graph. Every figure is computed deterministically from the coefficients you enter, never guessed, so the same polynomial always yields the same roots. The method and a worked example that reconciles exactly to the calculator default appear in full below for you to check.
The roots are the x values where the polynomial equals zero. For the default cubic x^3 - 6x^2 + 11x - 6 = 0, the three real roots are 1, 2 and 3, since it factors as (x-1)(x-2)(x-3).
Formula
a x^4 + b x^3 + c x^2 + d x + e = 0
Leave a (and b) at 0 for a cubic or quadratic
Real roots are found by sign-change scanning and bisection
The calculator treats the inputs as a polynomial of degree up to four. It scans for sign changes, which bracket the real roots, and uses repeated bisection to locate each one to a very small tolerance.
Worked example
Solve x^3 minus 6x^2 plus 11x minus 6 equals 0, entering 0, 1, -6, 11 and -6 as the coefficients.
- The cubic factors as (x-1)(x-2)(x-3)
- Each factor is zero at x = 1, x = 2 and x = 3
- Check x = 2: 8 - 24 + 22 - 6 = 0, confirming a root
- The three real roots are 1, 2 and 3
These are the calculator's default inputs, so the roots above match the widget exactly.
Roots of a Polynomial Calculator: frequently asked questions
What degree polynomials does this solve?
Up to degree four. Set the coefficient of x^4 to zero for a cubic, and both x^4 and x^3 to zero for a quadratic. The tool handles each case automatically.
How does it find the roots?
It evaluates the polynomial across a wide range, finds every interval where the sign changes (which must contain a real root), and refines each interval with bisection until the root is located precisely.
Does it show complex roots?
No. This calculator reports real roots only, the values where the graph crosses the horizontal axis. Complex roots occur in conjugate pairs and are not displayed.
Why are some entered values ignored?
If the leading coefficients are zero the polynomial is of a lower degree, so the tool drops those terms and solves the reduced polynomial.
Are the results exact?
They are computed to a very small tolerance, accurate to the decimals shown. For integer roots like the default, the answer is exact.
Official sources
- Polynomial root-finding and numerical methods reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.