Polynomial Zeros Calculator
Find real and complex polynomial zeros from coefficient lists up to degree eight, inspect residual errors, and view roots on a responsive complex-plane plot.
Polynomial Zeros Calculator inputs
Change the calculator-specific fields below. The result, checks, and visual update from this calculator’s own formula engine.
Use zero for missing powers. Example: x³ − 6x² + 11x − 6 is entered as 1 -6 11 -6.
Polynomial Zeros Calculator results
How the polynomial zeros calculator works
This page is built for the exact search intent represented by polynomial zeros calculator. It exposes the governing assumptions and supporting quantities instead of returning an unexplained headline number.
Coefficient-list input
Enter coefficients in descending power order and include zero placeholders for missing terms. This avoids ambiguous expression parsing and gives the numerical solver a precise degree and coefficient vector.
Real and complex roots
A degree-n polynomial has n complex roots when multiplicity is counted. The calculator uses a simultaneous complex-root iteration and reports both real and nonreal roots. A root with a tiny imaginary component is cleaned to a real value only when the component is below the displayed numerical tolerance.
Residual checking
Every reported zero is substituted back into the original polynomial. The residual |P(x)| shows how closely the numeric root satisfies the equation. Repeated roots and poorly scaled coefficients can converge more slowly, so the residual is more informative than the number of displayed digits.
Reliable-use checklist
Confirm that the selected mode, units, signs, dates, rates, and definitions match the original problem. Retain the detailed output and supporting cards so the answer can be reproduced. Test a known example before relying on a custom case, and use current official or professional guidance when taxes, benefits, safety, contracts, or regulated decisions are involved.
The polynomial zeros calculator runs locally in the browser. It does not transmit entered values, create an account, or retain a calculation history. Full internal precision is kept during calculation and display rounding is applied only to readable outputs.
Polynomial Zeros Calculator questions
Why are there complex roots?
The fundamental theorem of algebra counts roots over the complex numbers, even when a polynomial has few or no real crossings.
How do I enter a missing x term?
Insert a zero coefficient in that position.
Are the roots exact radicals?
The results are numerical approximations with residual checks, not symbolic radical forms.