Calculator Input
Example Data Table
| Input | Power | First Few Terms | Total Terms |
|---|---|---|---|
| (x + y + z)2 | 2 | x2 + 2xy + 2xz + y2 + 2yz + z2 | 6 |
| (2x + 3y + z)3 | 3 | 8x3 + 36x2y + 12x2z ... | 10 |
| (x2 + y + 4z)4 | 4 | x8 + 4x6y + 16x6z ... | 15 |
Formula Used
For a trinomial (A + B + C)n, each expanded term is:
n! / (i! j! k!) × Ai × Bj × Ck
Here, i + j + k = n. The calculator loops through every valid combination of i, j, and k. It multiplies the trinomial coefficient by the selected term coefficients. Then it applies the variable powers entered in the form.
How to Use This Calculator
- Enter the coefficient for each of the three terms.
- Enter each variable name, such as x, y, or z.
- Enter the power attached to each variable.
- Enter the whole expansion power from 0 to 20.
- Choose decimal precision and term ordering.
- Press the expand button to view the result.
- Use CSV or PDF buttons to save your work.
About Trinomial Expansion
A Clear Algebra Tool
A trinomial expansion rewrites a power of three added terms as a sum of many separate terms. It is common in algebra, probability, statistics, and symbolic modeling. Manual expansion can become slow when the power increases. This calculator helps by listing each valid term and its coefficient.
Why the Multinomial Rule Matters
The calculator uses the multinomial theorem. This theorem extends the binomial theorem to three or more terms. Each term depends on three counters named i, j, and k. These counters must always add up to the selected expansion power. That rule keeps the expansion complete and avoids repeated terms.
Advanced Input Control
You can enter different coefficients, variables, and variable powers. This supports simple cases like (x + y + z)3. It also supports richer cases like (2x2 + 3y + 5z)4. Negative coefficients are accepted. Decimal coefficients are accepted too. The precision setting controls the displayed rounding.
Useful for Learning and Checking
Students can use the detailed table to understand where each term comes from. Teachers can use the CSV export to prepare examples. The PDF export is useful for homework notes and classroom review. The coefficient sum also gives a quick check. It equals the expansion evaluated when all variables are set to one.
Better Than a Short Answer
A final expression alone can hide the process. This page shows the source expression, term count, multinomial coefficient, numeric coefficient, variable part, and final term. That makes errors easier to find. It also makes the method easier to remember.
FAQs
1. What is a trinomial expansion?
A trinomial expansion rewrites a power of three added terms into a full sum of individual algebraic terms.
2. Which formula does this calculator use?
It uses the multinomial theorem. Each term uses n! divided by i!, j!, and k!, where i + j + k equals n.
3. Can I use negative coefficients?
Yes. Negative and decimal coefficients are allowed. The result keeps the correct signs in the expanded expression.
4. Can variables have powers?
Yes. You can enter a separate power for each variable. The calculator multiplies that power by each term counter.
5. Why is the maximum expansion power limited?
Higher powers create many terms. The limit keeps the page fast, readable, and practical for normal algebra work.
6. What does coefficient sum mean?
It is the expanded expression evaluated when all variables equal one. It is a quick way to check coefficient totals.
7. What is included in the CSV file?
The CSV file includes the source expression, expanded result, counters, multinomial coefficients, numeric coefficients, variables, and terms.
8. Is the PDF file complete?
The PDF includes the main result and a term summary. For very large expansions, use the CSV file for the full table.