Polynomial Calculator
Evaluate any polynomial at any x by Horner’s method — the steps printed, the cost counted, and the factor theorem checked live when a value lands on zero.
Polynomial Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- One evaluation per run — no tables of values
- No factoring or root listing (the Equation Solver owns roots numerically)
In short: Coefficients 1, −3, 2 (that is x² − 3x + 2) at x = 4: Horner nests it as ((1)·4 − 3)·4 + 2 = 6, two multiplications instead of a table of powers. The degree is 2, the leading coefficient 1, the constant 2. And x = 1 is spotted live: f(1) = 0, so (x − 1) is a factor — the factor theorem working in front of you.
Formula
f(x) = (((c₀)x + c₁)x + c₂)x + …
n multiplications, never a power table
Horner, 1819 — the same nesting your calculator still uses.
Worked Example
- Parse. coefficients read highest degree first: 1, −3, 2 is 1x² + (−3)x + 2. Zeros are placeholders — 2, 0, −4, 5 owns an x³ with no x².
- Nest. Horner: start from the leading coefficient, multiply by x, add the next, repeat. Each coefficient is touched exactly once.
- Inspect. degree, leading coefficient and constant printed; if f lands on zero, the factor theorem is invoked by name — (x − r) divides.
The nesting is the whole idea: x² − 3x + 2 at x = 4 is ((1)·4 − 3)·4 + 2 — two multiplies and two adds. Naive evaluation computes 4² from scratch and does more work for the same answer; at degree 20 the gap is 20 multiplications versus 210.
Strengths & Limits Of This Model
Where this engine is strong
- Every Horner step printed — the nesting is the lesson, not just the number
- The factor theorem fires live when f(r) = 0
Where it stops
- No synthetic division output
- No rational-root candidate list
Practical Use Cases
Graphing prep
plot points fast — each point costs one Horner pass
Root hunting by hand
test candidate roots; a zero hands you a factor
Numerics
the form numerical code actually uses, printed step by step
Methodology & Editorial Standards
Coefficients parsed from the comma list (highest degree first, any reals, zeros as placeholders). Horner’s nesting evaluated left to right with each step recorded: n multiplications for degree n. Degree, leading coefficient, constant and the standard form are printed; f(x) = 0 triggers the factor theorem by name. The all-zero list is refused as the zero polynomial.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Polynomial Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What is Horner’s method actually doing?
Rewriting the polynomial as nested multiplication: x² − 3x + 2 is ((1)x − 3)x + 2. You walk the coefficients once, multiply by x and add the next one each step. Same value as the textbook form, but the powers of x are never computed separately — they accumulate inside the nesting.
Why is it faster than evaluating powers directly?
Count the multiplications. Direct evaluation at degree n needs 1 + 2 + … + n = n(n+1)/2 multiplications for the powers alone; Horner needs exactly n. At degree 4 that is 20 versus 4; at degree 20 it is 210 versus 20. Same answer, an order of magnitude less work — which is why every serious numerical library evaluates polynomials this way.
Why list coefficients highest degree first?
Because Horner eats them in that order — leading coefficient first, constant last. 1, −3, 2 means 1x² + (−3)x + 2. To hold a missing degree, use a zero: 2, 0, −4, 5 is 2x³ − 4x + 5 with the x² slot empty but counted. The length of the list IS the degree plus one.
What is the factor theorem doing on this page?
Closing the loop between values and factors: f(r) = 0 exactly when (x − r) divides the polynomial. The page watches every evaluation — the moment f lands on zero it says so and names the factor. For the default polynomial, x = 1 gives f(1) = 0, so (x − 1) is a factor — which is also why x = 1 is one of the preset drives.
Can coefficients be decimals or negatives?
Any real numbers work — Horner never assumes integers. Negative coefficients are entered with a minus (1, -3, 2), decimals evaluate exactly as doubles, and the constant-only polynomial 5 evaluates to 5 at every x, degree zero and proud of it.
What does a zero coefficient do?
It holds the degree’s place. 2, 0, -4, 5 has degree 3 even though x² is absent — the zero rides through Horner as “multiply by x, add nothing”. Dropping it would silently change the polynomial: 2, -4, 5 is a different, quadratic animal.
What is the zero polynomial?
Every coefficient zero: degree undefined, value zero at every x. It is the one polynomial with no degree and no leading coefficient, and the page refuses to invent either — it says “zero polynomial” and evaluates to 0 honestly.
Does the page factor polynomials for me?
No — and the limit is stated, not hidden. Full factoring is root-finding in costume, and root-finding has its own page (the Equation Solver). This page evaluates honestly, names the method, counts the cost, and invokes the factor theorem when a value happens to hit zero — a window, not a claim.
Why print the standard form if I just typed it?
Because the input is a LIST and the mind wants the POLYNOMIAL: 1, -3, 2 read back as x² − 3x + 2 catches transcription slips instantly — wrong order, a dropped zero, a missing minus. The shape card is the page’s way of showing you what it thinks you gave it, before the number matters.