Calculator inputs
Example data table
| Mode | Input | Target | Output coefficient |
|---|---|---|---|
| Standard binomial | (2 + 5x)6 | x3 | 4000 |
| Shifted binomial | (2x + 3x2)5 | x7 | 2160 |
| Trinomial | (1 + 2x + x2)4 | x4 | 70 |
| Custom polynomial power | (1 + 2x + x2)3 | x2 | 15 |
Formula used
1) Standard binomial coefficient
For (a + bx)^n, the coefficient of x^k is C(n,k) · an-k · bk, provided 0 ≤ k ≤ n.
2) Shifted binomial coefficient
For (a x^p + b x^q)^n, pick the second term exactly i times. The exponent becomes p(n-i) + qi, and the related coefficient becomes C(n,i) · an-i · bi.
3) Trinomial coefficient
For (a + bx + cx²)^n, sum every valid triple (i, j, l) such that i + j + l = n and j + 2l = k. Each contribution is n!/(i!j!l!) · ai · bj · cl.
4) Custom polynomial power
For P(x)^n, the calculator multiplies the polynomial by itself repeatedly. This convolution method adds coefficients whose exponents combine to the requested power.
How to use this calculator
- Choose a method that matches your algebraic expression.
- Enter the constants, powers, and the target exponent
k. - Use comma-separated coefficients for the custom polynomial mode.
- Click Find coefficient to generate the answer above the form.
- Review the result card, steps, table, and graph to verify the pattern.
- Use the CSV and PDF buttons to export the current output.
FAQs
1) What does this calculator actually find?
It finds the numerical coefficient attached to a chosen power of x in several expression types, including standard binomials, shifted binomials, trinomials, and custom polynomial powers.
2) Can it handle exponents that do not appear?
Yes. If the requested power cannot occur in the selected expansion, the calculator returns zero and explains why through the computation steps.
3) How should I enter a custom polynomial?
Enter coefficients in ascending order of powers. For example, 3, -1, 4 means 3 - x + 4x².
4) Why are there four different modes?
Each mode matches a different algebra structure. Using the right mode gives cleaner formulas, more meaningful steps, and a distribution graph that matches the chosen expansion.
5) What does the graph show?
The Plotly graph shows every nonzero coefficient produced by the current expression, plotted against its exponent. It helps you see symmetry, spread, and dominant terms.
6) Are decimal inputs allowed?
Yes. Constants and polynomial coefficients can be decimal values. Integer powers are still required so the expression remains a polynomial-style expansion.
7) Why are some mode limits smaller?
Higher-degree expansions grow quickly in size. The limits keep the calculator responsive while still supporting advanced classroom, homework, and verification use cases.
8) What is included in the exported files?
The exports include the main result summary and the coefficient distribution table, making it easier to save, share, print, or reuse the calculation.