Polynomial subtraction calculator
Subtract polynomials column by column: 4x^3 − 2x + 7 minus x^3 + 5x^2 − 3 gives 3x^3 − 5x^2 − 2x + 10, with every power lined up so you can check each step. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Minuend
- 4x^3 - 2x + 7
- Subtrahend
- x^3 + 5x^2 - 3
- Expression
- 3x^3 - 5x^2 - 2x + 10
- Degree
- 3
This polynomial subtraction calculator turns two expressions in x into one simplified difference. Enter the polynomial you are subtracting from as the minuend and the polynomial being removed as the subtrahend. The calculator groups repeated powers, aligns every like term, changes the effective sign of the second polynomial, and combines the coefficients. It returns a readable expression in descending-degree standard form, structured nonzero terms, the resulting degree, and a complete alignment table that keeps canceled terms visible. Decimal coefficients, missing powers, and coefficients of one are handled directly.
How to subtract polynomials by aligning like terms
Polynomial subtraction starts by identifying terms with the same variable raised to the same exponent. A cubic term can combine only with another cubic term, a square term only with another square term, and a constant only with another constant. Write both polynomials in descending-degree order and insert a zero placeholder wherever one polynomial has no matching power.
Then subtract the coefficient in the second row from the coefficient in the first row. This calculator performs exactly that alignment even when the input terms arrive out of order or the same power appears more than once. For example, subtracting x^3 + 5x^2 - 3 from 4x^3 - 2x + 7 aligns powers three, two, one, and zero.
Their coefficient differences are three, negative five, negative two, and ten, producing 3x^3 - 5x^2 - 2x + 10. The alignment table in the response retains all four rows, including any row whose difference becomes zero. The compact terms list and expression omit those zeros because standard form contains only nonzero terms.
This separation makes the answer useful both as a final result and as evidence of the coefficient arithmetic used to reach it. If every coefficient cancels, the expression is zero and the reported degree is negative one, a conventional machine-friendly representation of the zero polynomial.
Entering expressions and reading the result
Provide two required text fields: minuend and subtrahend. The minuend is the polynomial from which the other polynomial is taken; operand order matters because A minus B generally differs from B minus A. Write the variable as x or X.
Accepted terms include 7, x, -x, 3x, 3*x, 2.5x^4, and decimal constants such as .75. Exponents must be explicit nonnegative integers when present, so x^0 is accepted as a constant while x^-1 is rejected. Spaces are optional, and the Unicode minus sign is normalized for convenient pasted input.
Parentheses, fraction notation such as 1/2, multiple variables, and implicit products of expressions are outside this focused parser. The response expression is already sorted from highest exponent to the constant term. Its terms array contains objects with exponent and coefficient fields for nonzero results.
The degree equals the highest remaining exponent, or negative one when the result is zero.
Checking signs, cancellation, and practical uses
The most common polynomial-subtraction mistake is distributing the subtraction sign incompletely. Every term of the subtrahend is subtracted, including terms that already carry a negative sign. In coefficient language, the calculation is direct: for each exponent, compute the minuend coefficient minus the subtrahend coefficient.
Thus subtracting negative three at the constant position adds three to the first constant. A reliable manual check is to add the reported difference back to the subtrahend; after like terms are combined, that sum should reproduce the minuend. You can also reverse the operands and confirm that every nonzero coefficient changes sign.
The structured alignment is useful in tutoring software because it can display missing terms as zeros without polluting the final standard-form answer. It also handles cancellation cleanly: subtracting a polynomial from itself returns expression zero, an empty nonzero terms list, and degree negative one while the alignment still shows every canceled power. Results use finite JavaScript numbers normalized to stable precision, so this tool is intended for ordinary decimal classroom coefficients rather than exact rational arithmetic with enormous integers.
What you can do with it
Check algebra homework
Compare a hand-computed difference with the standard-form expression and inspect each aligned coefficient when a sign error appears.
Create worked answer keys
Use the alignment rows to present the minuend, subtrahend, and coefficient difference for every power in a classroom exercise.
FAQ
Which polynomial goes in each field?
Put the polynomial you start with in minuend and the polynomial being taken away in subtrahend. The operation is minuend minus subtrahend.
Does the calculator combine repeated like terms?
Yes. Terms with the same exponent are combined within each input before the two polynomials are subtracted.
Can the inputs have missing powers or different degrees?
Yes. A missing power is aligned with a zero coefficient, and the final expression is sorted by descending exponent.
What happens when all terms cancel?
The expression is 0, the nonzero terms list is empty, and the degree is -1. Alignment rows remain available to show the cancellation.
Does the order of the two polynomials matter?
Yes. A minus B is generally not the same as B minus A: swapping them flips the sign of every coefficient in the result. Put the polynomial you subtract from first and the one being removed second.
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-subtract \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"minuend":"4x^3 - 2x + 7","subtrahend":"x^3 + 5x^2 - 3"}'const res = await fetch("https://api.kit.forhosting.com/algebra/polynomial-subtract", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"minuend": "4x^3 - 2x + 7",
"subtrahend": "x^3 + 5x^2 - 3"
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/polynomial-subtract",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"minuend": "4x^3 - 2x + 7",
"subtrahend": "x^3 + 5x^2 - 3"
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/polynomial-subtract", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"minuend":"4x^3 - 2x + 7","subtrahend":"x^3 + 5x^2 - 3"}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"minuend":"4x^3 - 2x + 7","subtrahend":"x^3 + 5x^2 - 3"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/polynomial-subtract", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"minuend": "4x^3 - 2x + 7",
"subtrahend": "x^3 + 5x^2 - 3"
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.polynomial_subtract",
"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 same deterministic function powers browser and API results, with successful API requests priced at $0.002 per item.
Entering expressions and reading the result
The aligned_terms array shows the minuend coefficient, subtrahend coefficient, and difference coefficient at every power found in either input. That structure is especially convenient for rendering a worked vertical subtraction or checking one column at a time. Invalid syntax produces an invalid_input response with a message that identifies the relevant field or term instead of silently guessing what the writer meant.
Checking signs, cancellation, and practical uses
The implementation is pure and makes no network request, and a successful API calculation costs the published base price of $0.002 per item.
Integration use cases
Test symbolic interfaces
Generate stable structured terms and expressions for regression fixtures without depending on a remote computer algebra system.
Technical questions
What does polynomial subtraction cost?
The browser calculation is free. A successful API request costs $0.002 per item.