Polynomial addition calculator
Combine like terms in one step: 1 + 2x + 3x² plus 3 + x² gives 4 + 2x + 4x², and subtracting instead gives −2 + 2x + 2x². The degree of each polynomial is shown too. Free in your browser.
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Runs in your browser. Free, unlimited — your data never leaves this page.
Polynomial addition and subtraction are the first ring operations every algebra student learns after combining like terms. This calculator takes two univariate polynomials as dense coefficient lists—constant term first, then x, then x squared, and so on—adds or subtracts them coefficient-wise, and returns the simplified list with trailing zeros removed. You also get the degree of each input and of the result, plus human-readable expressions for copy-paste into homework or notebooks.
How polynomial addition and subtraction work on coefficient lists
A dense univariate polynomial is completely described by the ordered list of its coefficients for ascending powers of x. The list p equals p zero, p one, p two, and so on means p of x equals p zero plus p one times x plus p two times x squared and higher terms until the last index. To add two such polynomials you align them on the constant term, pad the shorter list with zeros on the high-degree end, and add matching positions.
Subtraction is the same alignment with a minus sign on every coefficient of the second polynomial. After the pointwise sum or difference, any trailing zeros at the highest degrees are dropped so the length of the result equals one plus the true degree, except for the zero polynomial which is represented as a single zero with conventional degree negative one. That simplification is exactly what textbook “combine like terms and write in standard form” asks for, and it is what this engine returns as the result array.
The algorithm is deterministic linear time in the length of the longer list: no factorization, no root finding, no floating-point iteration beyond ordinary addition of finite numbers. Coefficients must be finite; strings that parse as numbers are accepted for convenience in text fields, but objects, booleans used as flags, and non-numeric tokens are rejected. Because addition is associative and commutative over the reals (or rationals you embed as floats), reordering p and q under the add operation yields the same simplified result, while subtraction reverses when operands swap.
Where polynomial add and subtract show up in real work
Secondary-school and college algebra constantly assign combine-like-terms drills, standard-form rewrites, and “add these two polynomials” exercises; a single trusted oracle that returns both the coefficient vector and a readable expression speeds grading and demo setup. Computer-algebra homework often begins with dense addition before students meet sparse monomial maps or multivariate rings, 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 polynomial addition under generating functions, signal FIR kernels, and path-counting recurrences where two coefficient arrays must be summed before further transforms. Data and science notebooks combine polynomial trend models or characteristic polynomials from adjacent pipeline stages and need a deterministic strip of leading zeros rather than a silent type-coercion bug.
If you later need polynomial multiplication, division with remainder, evaluation at a point, or root finding for quadratics and cubics, treat those as separate search intents and landings rather than overloading this polynomial-addition calculator page.
What you can do with it
Homework combine-like-terms keys
Generate simplified coefficient lists and printable expressions for textbook p±q drills so teachers and auto-graders share one ground truth.
Generating-function and FIR pipelines
Sum or difference dense coefficient arrays in contest code and signal notebooks without hand-rolling pad-and-zip logic each time.
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: [1, 2, 3] means 1 + 2x + 3x².
What happens with different degrees?
The shorter list is treated as padded with zeros on the high-degree end. After the operation, trailing zeros are stripped so the result degree is exact.
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.
What happens when the two polynomials cancel out?
You get the zero polynomial, shown as a single 0. For example, subtracting 1 + x from 1 + x leaves nothing, and any zeros left at the highest powers are always removed from the result.
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-add \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"p":[1,2,3],"q":[3,0,1]}'const res = await fetch("https://api.kit.forhosting.com/algebra/polynomial-add", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"p": [
1,
2,
3
],
"q": [
3,
0,
1
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/polynomial-add",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"p": [
1,
2,
3
],
"q": [
3,
0,
1
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/polynomial-add", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"p":[1,2,3],"q":[3,0,1]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"p":[1,2,3],"q":[3,0,1]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/polynomial-add", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"p": [
1,
2,
3
],
"q": [
3,
0,
1
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.polynomial_add",
"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
Non-numeric coefficients, empty lists, and NaN or Infinity values 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 addition and subtraction work on coefficient lists
Identical inputs always produce identical JSON: there is no random seed, no clock, and no network call on the path.
What you send and what you get back
Send two required arrays under the fields p and q. Each entry must be a finite number representing the coefficient of x to the power of its index, with index zero the constant term. Optionally send op as add or sub; the default is add. Classic checks: p equals one, two, three and q equals three, zero, one with add yields four, two, four which is four plus two x plus four x squared; the same lists with sub yield negative two, two, two. Cancellation examples such as p equals one, one and q equals one, one with sub yield the zero polynomial as a single zero. Short aliases such as plus, plus sign, subtract, and minus are accepted and normalized. Missing arrays, empty arrays, non-numeric coefficients, NaN, and Infinity fail with invalid_input and are not charged. On success the response echoes cleaned p and q with their own trailing zeros stripped, the normalized op, the simplified result array, degrees for result and both inputs, printable expression strings for all three polynomials, and an operation_label of p plus q or p minus q for UI badges. 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 per polynomial so accidental multi-megabyte payloads cannot stall a shared worker.
Where polynomial add and subtract show 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 add/sub contract from a free browser form and from regression suites so demos never diverge from billed API output.
Technical questions
How do I subtract instead of add?
Set op to "sub" (aliases: subtract, minus, -). The default op is "add". The response operation_label reflects p + q or p - q.