Polynomial even/odd symmetry checker
Enter 1, 0, -3, 0, 2 and you learn that x^4 - 3x^2 + 2 is even, because every nonzero term sits on an even power. The checker reads the exponents directly, so you never expand f(-x) by hand. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Coefficients
- 1, 0, -3, 0, 2
- Degree
- 4
- Polynomial
- x^4-3x^2+2
- Symmetry
- even
- Even exponents
- 4, 2, 0
Even and odd functions are the first symmetry test many calculus students meet: f is even when f of negative x equals f of x, and odd when f of negative x equals the negative of f of x. For a univariate polynomial the test collapses to a simple parity rule on the exponents that actually appear. Only even powers can survive in an even polynomial; only odd powers can survive in an odd polynomial; mixed powers mean neither; the zero polynomial is both.
What even and odd mean for a polynomial
A real function f is even when substituting negative x leaves the graph unchanged under reflection across the y-axis, which is the identity f of negative x equals f of x for every x in the domain. It is odd when the graph has origin symmetry, which is the identity f of negative x equals the negative of f of x. Those two identities are not exclusive in a vacuous sense: the zero function satisfies both, because zero equals its own negative.
For a polynomial written as a sum of a sub k times x to the k, the substitution x maps to negative x multiplies every term by negative one raised to the power k. Terms with even k are unchanged, so they belong in even functions; terms with odd k flip sign, so they belong in odd functions.
Therefore a polynomial is even exactly when every nonzero coefficient sits on an even exponent, odd exactly when every nonzero coefficient sits on an odd exponent, both when every coefficient is zero, and neither when the support of the polynomial contains at least one even and one odd power. This checker does not need to evaluate f at sample points: it reads the coefficient vector once, records the parities of the nonzero slots, and emits the textbook classification.
That makes the result exact for finite floating coefficients after a small cleanup of near-zero noise, and it matches how instructors mark paper answers when they ask students to inspect exponents rather than expand f of negative x by hand.
Who benefits from a quick parity check
Classrooms and tutoring platforms use a polynomial even odd symmetry checker so learners can confirm whether a coefficient list is even or odd without expanding f of negative x by hand and risking sign errors on every term. Exam generators and LMS plugins need a deterministic oracle that marks the zero polynomial as both and rejects empty input instead of silently treating it as a constant zero that a buggy grader might mishandle.
Computer-algebra front ends and symbolic notebooks often want a cheap pre-filter before heavier simplifications: if a polynomial is already pure even or pure odd, half of the terms in an f of negative x expansion can be skipped. Failed validation does not consume balance.
What you can do with it
Classroom and homework checks
Let students submit coefficient lists and verify even, odd, neither, or both against the exponent-parity rule used in calculus textbooks.
Exam and LMS auto-grading
Mark even/odd classification items with a deterministic oracle that treats the zero polynomial as both and never invents mixed-parity labels.
CAS pre-filters and notebooks
Skip half of an f(-x) expansion when a polynomial is already pure even or pure odd before heavier symbolic work.
FAQ
What makes a polynomial even?
Only even powers have nonzero coefficients, so f(-x)=f(x). Example: x^4−3x^2+2 is even; x^3+x is odd.
Do I need to expand f(-x) myself?
No. The classification inspects exponent parity of nonzero terms only; it does not evaluate the polynomial at sample points.
How do I check if a polynomial is even or odd?
Enter the coefficients from the highest power down to the constant term. For x^4 - 3x^2 + 2, type 1, 0, -3, 0, 2: only even powers have nonzero coefficients, so the polynomial is even.
Can a polynomial be both even and odd?
Only the zero polynomial. Every coefficient is zero, and zero equals its own negative, so it passes both mirror tests at once. Any polynomial with a nonzero term is even, odd or neither.
Why is x + 1 neither even nor odd?
It mixes an odd power, x, with an even power, the constant term. Even symmetry needs only even powers and odd symmetry needs only odd powers, so a mix of both fails each test.
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/calculus/polynomial-symmetry \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"coefficients":[1,0,-3,0,2]}'const res = await fetch("https://api.kit.forhosting.com/calculus/polynomial-symmetry", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"coefficients": [
1,
0,
-3,
0,
2
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/polynomial-symmetry",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"coefficients": [
1,
0,
-3,
0,
2
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/polynomial-symmetry", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"coefficients":[1,0,-3,0,2]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"coefficients":[1,0,-3,0,2]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/polynomial-symmetry", 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": [
1,
0,
-3,
0,
2
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.polynomial_symmetry",
"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
This polynomial even/odd symmetry checker accepts coefficients highest degree first, lists the even and odd exponents with nonzero coefficients, and returns a clear symmetry label plus boolean flags.
Input contract, response shape, and error behaviour
The entry at index zero is the leading coefficient of x to the power n where n is length minus one, and the last entry is the constant term. Zero entries are allowed and simply mark missing powers; they do not change the parity sets. On success the body echoes the cleaned coefficients, reports the effective degree as the highest power with a nonzero coefficient (zero for the zero polynomial), reprints a compact polynomial string for display, and sets symmetry to even, odd, neither, or both. Example: coefficients one, zero, negative three, zero, two represent x to the fourth minus three x squared plus two, which is even; coefficients one, zero, one represent x cubed plus x, which is odd; coefficients one, one represent x plus one, which is neither. POST a JSON body with one required field: coefficients, a non-empty array of finite numbers ordered highest degree first. Aliases coeffs and poly are accepted for the same array. Empty arrays, non-arrays, missing fields, and non-finite entries fail with invalid_input and are not billed. Degree is capped for safety so a runaway payload cannot force an oversized scan. Boolean fields is_even and is_odd mirror the classical identities, so clients that prefer flags over a string enum do not need extra branching. Arrays even_exponents and odd_exponents list the powers that carried nonzero coefficients, sorted descending. nonzero_term_count is their combined length, and formula restates the parity rule in one line for UI footers.
Where a polynomial symmetry API earns its keep
Engineering curricula that introduce Fourier series, odd and even extensions, or integration over symmetric intervals rely on the same vocabulary, so a tiny parse API is enough to wire interactive demos without standing up a full computer-algebra system.
Technical questions
How do I check if a polynomial is even or odd?
Send coefficients highest degree first. The API lists even and odd exponents with nonzero coefficients and returns symmetry even, odd, neither, or both (zero polynomial).
What about the zero polynomial?
Every coefficient is zero, so f(-x)=0=f(x)=-f(x). The checker returns symmetry both with is_even and is_odd true.
What does the polynomial symmetry checker cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.