Evaluate a polynomial from descending-order coefficients at a chosen x value. The calculation trail makes the reported polynomial value easier to reproduce.
A coefficient list [aₙ,…,a₀] represents aₙxⁿ+…+a₀. Horner’s method nests the expression to reduce the number of multiplications.
For this page, polynomial evaluator is interpreted under the stated convention and input order.
Evaluation tests roots, fills function tables, supports interpolation, and checks polynomial identities numerically.
Zero coefficients must remain in the list to preserve missing powers. Coefficients are entered from highest degree to constant term. If the Polynomial Evaluator assumptions do not fit, consider multiply polynomials.
Compare the result with the worked example's scale before relying on the reported polynomial value.
Coefficients 2,−3,0,5 represent 2x³−3x²+5. At x=4, Horner’s method gives 85. This Polynomial Evaluator example can be compared with add polynomials.
Start with the leading coefficient, repeatedly multiply the running result by x, and add the next coefficient.
Source values may arrive in a different order from the form. Map them explicitly to coefficients, highest power first and value of x, normalize units, and retain enough precision for the next step after polynomial value.
Changing x can alter both magnitude and sign rapidly for high-degree polynomials. Watching this response separates a data-entry mistake from an unexpected but valid value.
If coefficients, highest power first and value of x are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Evaluation produces one function value; polynomial addition or multiplication constructs a new coefficient list. This page deliberately reports only the former interpretation.
Label the output as polynomial value in notes or tables. Store coefficients, highest power first and value of x beside it when the result will be reused in a later stage.
The polynomial 2x³−3x²+5 can be nested as ((2x−3)x+0)x+5. Reading coefficients in order makes the missing x term visible as zero. Evaluating both the expanded and nested forms at a simple value supplies an independent check on coefficient order.
They hold the place of missing powers.
Highest power first.
A nested evaluation requiring fewer operations.
Yes.