Calculator
Example data table
| Input | Value | Meaning |
|---|---|---|
| (a·x + b·y)^n | (2·x + 3·y)^4 | Expand a scaled binomial power |
| Mode | Full expansion | Compute every term from k=0..n |
| k | 2 | Pick a single term when needed |
| Variables | x, y | Symbols used in the final expression |
Quick output preview
Formula used
The binomial theorem expands a power of a sum: (p+q)^n = Σk=0..n C(n,k)·p^(n-k)·q^k. Here C(n,k) = n!/(k!(n-k)!) is the binomial coefficient.
This calculator uses p = a·x and q = b·y. So each term has coefficient C(n,k)·a^(n-k)·b^k, attached to x^(n-k)·y^k.
How to use this calculator
- Enter numbers for a and b (decimals or fractions).
- Enter an integer power n (0 to 60).
- Choose your variables (like x and y).
- Pick a mode: full expansion, a specific term, or the coefficient table.
- Click Calculate. Your result appears above the form.
- Use the download buttons to export CSV or PDF.
Coefficient Growth and Term Count
A binomial power (a·x + b·y)n produces n+1 terms, for k=0…n. Term count grows linearly, but coefficient size can grow fast: C(20,10)=184756, before scaling by a and b. For n=50, terms total 51, manageable in practice usually.
Symmetry and Peak Location
Coefficients satisfy C(n,k)=C(n,n−k). With equal scaling (a=b), the largest magnitude is near k≈n/2. Even n peaks at k=n/2, while odd n has twin peaks at (n−1)/2 and (n+1)/2. The chart often looks like a mirrored hill.
Scaling Effects of a and b
Term k has numeric coefficient C(n,k)·an−k·bk. If |b|>|a|, higher k terms are amplified. Example: with a=1, b=3, n=8, the b-power factor triples each step in k, shifting weight toward y-heavy terms.
kth Term Interpretation
The kth term corresponds to xn−k·yk. The same pattern appears in Bin(n,p) weights C(n,k)pk(1−p)n−k, where mid-range k values dominate for moderate p.
Numerical Precision and Rounding
Large n creates very large coefficients. The calculator computes C(n,k) with stable integer arithmetic, then applies scaling with powers. Increase precision when using small decimals (0.03) or when comparing adjacent terms.
Practical Uses in Modelling
Expansions support polynomial features, series approximations, and sensitivity checks such as (V+ΔV)n or (1+r)n. Exporting the coefficient table helps reuse values in spreadsheets, lessons, or reports.
FAQs
1) What does k mean in the binomial sum?
k is the term index from 0 to n. It controls the exponents: x^(n−k) and y^k, and selects the matching coefficient C(n,k)·a^(n−k)·b^k.
2) Can I use negative a or b values?
Yes. Negative inputs flip signs for terms where the negative base is raised to an odd power. The coefficient chart will show negative bars below the axis.
3) Why does the expansion have exactly n+1 terms?
Because k can take every integer value from 0 through n. Each k produces a unique pair of exponents (n−k, k), so the terms are distinct.
4) What is the difference between “kth term” and “coefficient table”?
“kth term” returns one selected term and its coefficient. “Coefficient table” lists all terms from k=0..n with C(n,k), scaled factors, and the final term expression.
5) How should I pick the precision setting?
Use higher precision for decimals and larger n. If your coefficients look rounded to the same value across neighboring k, increase precision to reveal meaningful differences.
6) What does the Plotly chart help me see?
It visualizes how coefficients change across k. You can quickly spot symmetry, peaks, sign changes, and whether scaling by a and b shifts weight toward x-heavy or y-heavy terms.