Polynomial Real Root Calculator
Find distinct real zeros of a degree-one through degree-six polynomial with derivative-based interval isolation and a residual table.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Use derivative roots and a Cauchy root bound to isolate real zeros, then apply bisection and repeated-root residual checks.
Solve aₙxⁿ+…+a₁x+a₀=0 for distinct real x
Worked example
Coefficients 1, −6, 11, −6 represent x³−6x²+11x−6=(x−1)(x−2)(x−3), giving distinct real roots 1, 2 and 3.
Assumptions and input details
At most degree 6 after leading zeros are removed. The all-zero polynomial is rejected. Coefficient-ratio root bounds above 1e8 are unsupported. Numerical repeated-root checks use a relative residual tolerance; roots closer than the stated scale tolerance can merge. Complex roots and exact multiplicity are not supplied. The plot is illustrative near the returned root region, not an exhaustive numerical proof.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Solve ax² + bx + c = 0, including real, repeated and complex roots. Inspect the discriminant, worked steps and the function graph.
Differentiate a polynomial of degree up to six, evaluate its slope at a point and find the tangent line. Enter coefficients and follow each power-rule term.
Divide one polynomial by another and follow each subtraction. See the exact quotient and remainder, the reconstruction identity and a substitution check.
Reserved advertising space.