Multiply rational expressions calculator
Multiply two polynomial fractions and get the reduced answer with its domain: (x + 1)/(x − 2) times (x − 2)/(x + 3) simplifies to (x + 1)/(x + 3), excluding the values 2 and −3. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Num1
- 1, 1
- Den1
- 1, -2
- Num2
- 1, -2
- Den2
- 1, 3
- Expression1
- (x + 1)/(x - 2)
- Expression2
- (x - 2)/(x + 3)
- Product unsimplified
- (x^2 - x - 2)/(x^2 + x - 6)
- Product
- (x + 1)/(x + 3)
Multiplying rational expressions is a core algebra skill: form the product of two polynomial fractions, factor when possible, cancel every common factor that appears in both the combined numerator and the combined denominator, then state the simplified product together with the values of the variable that must be excluded because they made an original denominator zero. Students, tutors, LMS authors, and automation scripts all need the same deterministic answer for textbook-style inputs without floating sign errors or forgotten domain holes. This multiply rational expressions calculator accepts four coefficient arrays—numerator and denominator of each factor—multiplies the polynomials, cancels their monic greatest common divisor with the Euclidean algorithm, and returns display strings, coefficient lists, the cancelled factor, and excluded real roots from the original denominators.
What multiplying rational expressions means and why cancel carefully
A rational expression is a ratio of polynomials, written n(x) over d(x) with d not the zero polynomial. The product of two such expressions is defined by multiplying numerators and multiplying denominators: (n1/d1) times (n2/d2) equals (n1 times n2) over (d1 times d2). That unsimplified product is algebraically correct but often not reduced.
Reduction means dividing numerator and denominator by every common polynomial factor—equivalently, by their monic greatest common divisor computed with the Euclidean algorithm on the product polynomials. After cancellation the remaining fraction is in lowest terms over the coefficient field used by the engine, and the denominator is normalized to be monic so clients see a stable leading one.
Critically, values that made either original denominator zero remain excluded from the domain of the original product even when those factors cancelled from the simplified form; textbooks call them removable discontinuities or holes, and graders still mark them as excluded. Linear denominators yield one excluded root each; quadratics yield zero, one, or two real roots via the quadratic formula; higher-degree denominators contribute rational roots found by the rational-root theorem when coefficients are near integers. Constant nonzero denominators contribute no finite excluded roots.
What you enter and what you get back
Example: num1 equals [1, 1] and den1 equals [1, -2] encode (x + 1) over (x − 2); num2 equals [1, -2] and den2 equals [1, 3] encode (x − 2) over (x + 3). Optional precision (integer zero through fifteen, default ten) controls decimal places on coefficients and roots. If den1 or den2 is the zero polynomial the task fails immediately.
Example: (x+1)/(x−2) times (x−2)/(x+3) simplifies to (x+1)/(x+3) with excluded values two and negative three because x equals two cancelled yet still leaves a hole.
Who checks rational products this way
Classrooms and tutoring platforms use a multiply rational expressions calculator so learners can check cancelled factors and excluded values for arbitrary coefficient lists without losing a common linear factor by hand. Exam-prep generators need a deterministic oracle that still lists cancelled zeros of original denominators instead of silently pretending the simplified denominator alone defines the domain. Engineering and science notebooks that compose transfer functions or rational approximations can multiply two proper or improper fractions of polynomials without importing a full computer-algebra system for a four-array product.
Failed validation does not consume balance. When your curriculum later asks for division of rational expressions, invert the second fraction and call the same multiply-and-cancel path; addition and subtraction require a common denominator and are out of scope for this product-focused tool.
What you can do with it
Classroom and homework checks
Students submit four coefficient arrays for two rational factors and verify the reduced product string plus excluded values against textbook methods.
Exam and LMS auto-grading
Mark multiply-and-simplify items with a deterministic oracle that still excludes cancelled denominator roots and never invents extra factors.
Notebooks and rational pipelines
Compose two polynomial fractions in scripts—transfer functions, filters, algebraic models—without a full computer-algebra library.
FAQ
How do I multiply two rational expressions with this calculator?
Send num1, den1, num2, and den2 as highest-degree-first coefficient arrays.
Why are cancelled factors still listed as excluded values?
Values that make an original denominator zero are outside the domain of the original product even after cancellation.
What coefficient order do the arrays use?
Highest degree first: [1, -3, 2] means x² − 3x + 2, and [1, -1] means x − 1. Empty arrays and the zero polynomial as a denominator are rejected.
What happens if a denominator is the zero polynomial?
A rational expression requires a non-zero denominator polynomial; den1 and den2 must each have at least one non-zero coefficient.
What does the product look like before it is simplified?
Multiplying straight across gives (x² − x − 2)/(x² + x − 6) for the example. Both parts share the factor x − 2, so it cancels and leaves (x + 1)/(x + 3).
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/multiply-rational-expressions \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"num1":[1,1],"den1":[1,-2],"num2":[1,-2],"den2":[1,3]}'const res = await fetch("https://api.kit.forhosting.com/algebra/multiply-rational-expressions", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"num1": [
1,
1
],
"den1": [
1,
-2
],
"num2": [
1,
-2
],
"den2": [
1,
3
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/multiply-rational-expressions",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"num1": [
1,
1
],
"den1": [
1,
-2
],
"num2": [
1,
-2
],
"den2": [
1,
3
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/multiply-rational-expressions", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"num1":[1,1],"den1":[1,-2],"num2":[1,-2],"den2":[1,3]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"num1":[1,1],"den1":[1,-2],"num2":[1,-2],"den2":[1,3]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/multiply-rational-expressions", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"num1": [
1,
1
],
"den1": [
1,
-2
],
"num2": [
1,
-2
],
"den2": [
1,
3
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.multiply_rational_expressions",
"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
What multiplying rational expressions means and why cancel carefully
This calculator therefore reports excluded_values from the original den1 and den2 roots, not only from the reduced denominator. Zero denominators are rejected up front as invalid_input because a rational expression is undefined when its denominator is the zero polynomial.
Input contract, response shape, and error behaviour
POST a JSON body with four required coefficient arrays in highest-degree-first order: num1 and den1 for the first rational expression, num2 and den2 for the second. Each coefficient may be a JSON number or a numeric string; missing fields, empty arrays, non-arrays, non-finite values, and degrees above thirty-two fail with invalid_input naming the field. On success the body echoes the four rounded inputs; expression1 and expression2 as pretty fractions; product_unsimplified as the raw product before cancel; product as the reduced fraction string; numerator and denominator coefficient arrays of the simplified result; numerator_poly and denominator_poly strings; cancelled_factor and cancelled_poly for the monic GCD that was divided out; cancelled true when a non-constant factor was removed; gcd_steps counting Euclidean iterations; excluded_values sorted, excluded_count, and excluded_note explaining domain holes; degree_numerator and degree_denominator; precision; method; and a short formula field. Pair the coefficients with your own factoring display if you need step-by-step binomial cards; this endpoint returns the exact reduced product and domain exclusions without multi-line prose proofs.
Where a multiply-rational-expressions API earns its keep
Keep requests within the degree and magnitude limits, pin the rounded product and excluded_values arrays in tests, and you get reproducible reduced fractions for homework, LMS plugins, and edge workers.
Technical questions
What does the multiply rational expressions calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.