Polynomial degree calculator
Read the degree of a polynomial straight off its coefficient list: 3, −2, 0 and 5 describe 3 − 2x + 5x³, a cubic of degree 3 whose leading coefficient is 5. Free in your browser.
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The degree of a polynomial is the highest power of the variable that appears with a nonzero coefficient. This calculator takes a univariate polynomial as a dense ascending coefficient list—constant term first, then x, then x squared, and so on—strips trailing zeros, and reports that highest power along with the leading coefficient and a human-readable expression.
How polynomial degree is defined on coefficient lists
A dense univariate polynomial is completely described by the ordered list of its coefficients for ascending powers of x. The list equals c zero, c one, c two, and so on means the polynomial equals c zero plus c one times x plus c two times x squared and higher terms until the last index. By definition the degree is the largest index i for which the coefficient of x to the i is nonzero.
That single number governs almost every structural fact students learn next: a nonzero polynomial of degree n has at most n roots counting multiplicity over an algebraically closed field, the product of two nonzero polynomials has degree equal to the sum of the degrees, and the leading term dominates asymptotic growth. When the input list ends with accidental zeros—common after addition, subtraction, or truncation—the true degree is smaller than length minus one, so this engine always scans from the high end and drops trailing near-zero entries before reporting.
Near-zero means absolute value below a tiny epsilon so floating-point dust does not invent a fake high term. The zero polynomial is special: every coefficient is zero after cleaning, there is no highest power with a nonzero coefficient, and textbooks conventionally write its degree as negative one or minus infinity. The algorithm is deterministic linear time in the length of the list: no factorization, no root finding, no random seed, no clock, and no network call.
Where polynomial degree shows up in real work
Secondary-school and college algebra constantly ask students to state the degree of a polynomial written in standard form or expanded from products; a single trusted oracle that returns both the integer degree and a readable expression speeds grading and demo setup. Computer-algebra homework often begins with dense degree checks before students meet sparse monomial maps or multivariate total degree, so an honest dense calculator matches the first unit of many syllabi without pretending to be a full CAS.
Programming contest problems and interview warm-ups hide degree under generating functions, recurrence characteristic polynomials, and filter order, where the highest nonzero coefficient decides buffer sizes and asymptotic complexity. Data and science notebooks combine polynomial trend models or characteristic polynomials from adjacent pipeline stages and need a deterministic strip of trailing zeros rather than a silent type-coercion bug that leaves a phantom high degree.
If you later need polynomial addition, multiplication, division with remainder, evaluation at a point, or root finding, treat those as separate search intents and landings rather than overloading this polynomial-degree calculator page.
What you can do with it
Homework degree drills
Generate ground-truth degree, leading coefficient, and printable expressions for textbook expand-and-state-the-degree exercises so teachers and auto-graders share one oracle.
Pipeline degree checks
After dense add, multiply, or truncation steps, confirm the true highest power before allocating buffers or claiming asymptotic order in contest and notebook code.
FAQ
What order are the coefficients in?
Ascending powers: index 0 is the constant term, index 1 is the coefficient of x, index 2 of x², and so on. Example: [3, -2, 0, 5] means 3 − 2x + 5x³, which has degree 3.
What is the degree of the zero polynomial?
A nonzero constant such as [7] has degree 0, which is different.
What if my list ends with zeros?
Trailing zeros are stripped before the degree is reported. [1, 2, 0, 0] simplifies to [1, 2] and degree 1 (linear).
What if a coefficient is not a number?
Every coefficient must be a finite number; NaN, Infinity, and non-numeric strings are rejected and not charged.
What does it cost?
The browser tool is free and runs locally.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/polynomial-degree \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"coefficients":[3,-2,0,5]}'const res = await fetch("https://api.kit.forhosting.com/algebra/polynomial-degree", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"coefficients": [
3,
-2,
0,
5
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/polynomial-degree",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"coefficients": [
3,
-2,
0,
5
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/polynomial-degree", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"coefficients":[3,-2,0,5]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"coefficients":[3,-2,0,5]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/polynomial-degree", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"coefficients": [
3,
-2,
0,
5
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.polynomial_degree",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
The zero polynomial is handled explicitly: every coefficient is zero, so the conventional degree is negative one and a dedicated boolean flag marks the case so callers never confuse “degree zero constant” with “no terms at all.” Non-numeric coefficients, empty lists, and NaN or Infinity fail with a clear invalid_input error so pipelines never silently treat garbage as zero. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
How polynomial degree is defined on coefficient lists
Here the numeric degree field is negative one and is_zero_polynomial is true so both human UIs and machine consumers can branch without parsing magic numbers alone. Coefficients must be finite; numeric strings are accepted for text fields, but objects, empty tokens, NaN, and Infinity are rejected with invalid_input.
What you send and what you get back
Send one required array under the field coefficients. Each entry must be a finite number representing the coefficient of x to the power of its index, with index zero the constant term. Short aliases coeffs and p are accepted for the same array. Missing arrays, empty arrays, non-numeric coefficients, NaN, and Infinity fail with invalid_input and are not charged. On success the response returns the cleaned coefficients with trailing zeros stripped, the integer degree, the boolean is_zero_polynomial, the leading_coefficient of the highest term (zero when the polynomial is the zero polynomial), the count of nonzero terms, a printable expression string, and a short degree_label such as constant, linear, quadratic, cubic, quartic, or a generic degree n phrase for higher degrees. Classic checks: coefficients three, negative two, zero, five means three minus two x plus five x cubed, so degree three with leading coefficient five; the list one, zero, zero is the constant one after stripping, degree zero; the list zero, zero, zero is the zero polynomial with degree negative one and is_zero_polynomial true. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance. List length is capped at ten thousand coefficients so accidental multi-megabyte payloads cannot stall a shared worker.
Where polynomial degree shows up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Curriculum widgets and API fixtures
Drive the same degree contract from a free browser form and from regression suites so demos never diverge from billed API output.