Expand Polynomials Calculator
Expand a product of polynomial factors — like (x + 2)(x − 3)^2 — into standard form. Exact rational coefficients (never rounded), plus a term-by-term breakdown.
Input
Use x as the variable, ^ for powers, and parentheses around each factor — e.g. (2x - 1)^3(x + 4). Coefficients can be integers, decimals, or exact fractions like 1/2.
Output
| Term (x^k) | Coefficient |
|---|---|
| No data yet | |
Guides
Multiplying out a product of polynomial factors by hand — especially once one of them is squared or cubed — means tracking a lot of terms without dropping a sign. This calculator expands the whole expression for you and returns it in standard form, using exact fraction arithmetic throughout so a coefficient like 1/2 or 1/3 never gets rounded to a decimal.
How to use it
- Enter an expression made of one or more polynomial factors in
x, each wrapped in parentheses — for example(x + 2)(x - 3)^2or(2x - 1)^3(x + 4). - Use
^for a power on a whole parenthesized factor ((x - 3)^2) or on a single term (x^3). - Coefficients can be plain integers, decimals, or exact fractions like
1/2. - Read off the expanded polynomial in standard form, its degree, and the term-by-term breakdown table.
What it handles
- Any number of factors multiplied together, each optionally raised to an integer power — not just the classic two-binomial FOIL case.
- Exact rational coefficients:
(1/2x + 1)(x - 2)expands to1/2x^2 - 2, not a rounded decimal. - A bare polynomial with no parentheses at all (e.g.
x^2 + 3x + 2) is accepted too — useful for double-checking that an expression is already in standard form. - Results up to degree 12; a larger request is rejected with an explanation rather than silently truncated.
FAQ
Can I use fractions or decimals as coefficients?
Yes — 1/2x, 0.5x, and x/2-style input all work (write the last one as 1/2x since division isn't parsed outside of a coefficient). Fractions are kept exact through every multiplication rather than converted to a rounded decimal.
What happens if terms cancel out?
The calculator drops any coefficient that comes out to exactly zero after expansion, so an expression like (x + 1)(x - 1) - x^2 correctly simplifies down to -1 with degree 0.
Is there a limit on how big an expression can be?
Yes — the expanded result is capped at degree 12 to keep the output readable. An expression that would expand past that (e.g. (x + 1)^20) returns an error explaining the limit instead of a truncated answer.
Related tools
To go the other direction and divide a polynomial back down, see the Synthetic Division Calculator. To solve a specific polynomial equation for its roots, see the Quadratic Formula Solver or Cubic Equation Solver.
Privacy
All calculations run entirely in your browser — the expression you enter is never sent to a server.
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/expand-polynomials-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"expression": "(x + 2)(x - 3)^2"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `expand-polynomials-calculator` tool (Expand Polynomials Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
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