Polynomial coefficients extractor
Paste a polynomial and get its coefficients in order, from the highest power down to the constant: 3x^2 + 2x − 5 gives 3, 2 and −5, and x^3 − 4x gives 1, 0, −4 and 0. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Expression
- 3x^2 + 2x - 5
- Variable
- x
- Degree
- 2
- Coefficients
- 3, 2, -5
- Standard form
- 3x^2+2x-5
Turning a written polynomial into a clean coefficient vector is the first step in almost every computer-algebra pipeline: root finders, synthetic division, Vieta checks, long division, and grading scripts all expect numbers in a fixed order, not free-form algebra text. Students still submit 3x^2 + 2x − 5; automation wants [3, 2, −5] from the highest power down to the constant. This polynomial coefficients extractor accepts a univariate expression string, combines like terms, and returns the ordered coefficient list together with degree, non-zero terms, and a compact standard-form rewrite.
What the coefficient list means and how ordering works
Once like terms are combined there is exactly one coefficient per power, including zeros for missing middle powers. Highest-first order matches synthetic division, many textbooks, and companion KIT polynomial tools that already document dividend coefficients that way. Degree is the largest power with a non-zero coefficient; the zero polynomial is reported as degree zero with coefficients [0].
Coefficients may be integers or decimals; near-integer floating noise is cleaned for stable machine-readable output. The response also lists only the non-zero terms with explicit power fields when you need a sparse view without scanning zeros, while the dense coefficients array remains the primary contract for numerical pipelines.
Input contract, accepted syntax, and error behaviour
Write the polynomial in expanded form: optional leading sign, terms joined by plus or minus, optional spaces, optional asterisk between coefficient and variable, and powers written as ^n or **n. Examples that succeed include 3x^2+2x-5, x^3 - 4x + 1, -2.5y^2+y, and the constant 7. Empty strings and missing expression fail the same way and are not billed.
Degree is capped for isolate safety so hostile megabyte strings cannot force huge dense vectors. Pair this extractor with downstream polynomial root, multiply, or divide capabilities by feeding coefficients straight into those array contracts.
Where a polynomial coefficient parser earns its keep
Classrooms and tutoring platforms use a polynomial coefficients extractor so learners can paste the expression they wrote on paper and see the ordered list used in synthetic division or the rational-root theorem without manually inserting zeros for missing powers. Exam generators and LMS auto-graders need a deterministic oracle that marks wrong a multi-variable paste or a trailing operator instead of silently dropping terms.
Engineering notebooks often receive polynomials from symbolic front-ends or OCR as text and must normalize them before numerical evaluation, filtering, or control-design steps that only accept coefficient arrays. Failed validation does not consume balance. Keep expressions expanded, pin the coefficient vectors in tests, and you get reproducible parsing for homework, LMS plugins, and edge workers without standing up a full computer-algebra system of your own.
What you can do with it
Classroom and homework checks
Let students paste 3x^2+2x-5 and verify the highest-first coefficient list [3,2,-5] before synthetic division or graphing steps.
Notebooks and numerical pipelines
Convert expression strings from OCR or CAS export into coefficient arrays for root finders, filters, and KIT polynomial APIs.
FAQ
How do I extract coefficients from a polynomial expression?
Send expression as a string such as 3x^2+2x-5.
Are missing powers filled with zero?
Yes. The dense coefficients array always has length degree+1. For x^3-4x you get [1,0,-4,0] so middle and constant zeros stay explicit.
Which variable letter is used?
If the expression contains one letter, that letter is used. Pure constants default to x. Pass optional variable to require a specific name.
Are like terms combined before the coefficients are listed?
Yes. Terms with the same power are added together first, so there is exactly one coefficient per power. You also get a compact standard form of the polynomial that you can copy back into your notes.
Which expressions are not accepted?
Expressions with two different letters, parentheses, division, an equals sign or scientific notation such as 1e3 are refused with a message that names the problem. Write the polynomial expanded, for example x^3 - 4x + 1.
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-coefficients \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"expression":"3x^2 + 2x - 5"}'const res = await fetch("https://api.kit.forhosting.com/algebra/polynomial-coefficients", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"expression": "3x^2 + 2x - 5"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/polynomial-coefficients",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"expression": "3x^2 + 2x - 5"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/polynomial-coefficients", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"expression":"3x^2 + 2x - 5"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"expression":"3x^2 + 2x - 5"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/polynomial-coefficients", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"expression": "3x^2 + 2x - 5"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.polynomial_coefficients",
"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
Empty input, multi-variable products, and unsupported characters fail with invalid_input instead of inventing a partial vector.
What the coefficient list means and how ordering works
A univariate polynomial is a finite sum of terms a_k · v^k where the exponent k is a non-negative integer and the coefficient a_k is a real number. This endpoint returns that sequence from the highest degree down to the constant: for 3x^2 + 2x − 5 the list is [3, 2, −5]; for x^3 − 4x the list is [1, 0, −4, 0] because the x^2 and constant positions are present as zeros so the vector length is always degree plus one. Ascending (constant-first) arrays used by some scientific libraries are simply the reverse; clients that need that layout reverse the array after the call.
Input contract, accepted syntax, and error behaviour
POST a JSON body with required string field expression (aliases poly, polynomial, expr, input). Optional variable (aliases var, v, letter) pins the letter when you want an explicit name; if omitted, a single letter found in the expression is used, or x for pure constants. Multi-variable expressions, products of two different letters in one term, parentheses, division, equals signs, and scientific notation such as 1e3 are rejected with invalid_input and a message that names the problem. On success the body echoes expression and variable, reports degree, coefficients as the dense highest-first array, terms as power and coefficient pairs for non-zero positions, and standard_form as a compact display string without spaces.
Integration use cases
Exam and LMS auto-grading
Normalize free-text polynomial answers into dense coefficient vectors and reject multi-variable or malformed expressions as invalid_input.
Technical questions
What syntax is not supported?
Parentheses, division, multi-variable products, equations, and scientific notation are rejected with invalid_input. Use expanded univariate form only.
What does the polynomial coefficients extractor cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.