Enter a, b, and n to expand instantly. Switch power order, see coefficients, and download tables. Learn the theorem, verify every term today.
This calculator expands (a·x + b)^n using the Binomial Theorem:
Here, C(n,k) = n! / (k!(n−k)!) is the binomial coefficient. Coefficients are computed exactly as integers.
| Input | Value | Sample output (first terms) |
|---|---|---|
| (a·x + b)n | (2·x + 3)5 | 243 + 810*x + 1080*x^2 + … |
| Order | Ascending | Terms listed from x^0 to x^n |
| Export | CSV / PDF | Coefficient table with binomial parts |
Coefficients follow C(n,k), which rises rapidly toward the center of the row. For n=20, the central value is C(20,10)=184,756; for n=50 it is C(50,25)=126,410,606,437,752. These counts explain why expansions become large even before applying a and b.
In practice, plotting C(n,k) alone shows the classic bell‑shaped profile, which is why coefficient magnitudes often resemble a discrete normal curve. The calculator’s table exposes this profile numerically and supports quick comparisons between different n values.
Each polynomial coefficient equals C(n,k)·a^k·b^(n−k). With a=2, b=3, n=10, the constant term is 3^10=59,049, while the highest‑power coefficient is 2^10=1,024. Middle terms can exceed both because they multiply sizable C(n,k) with mixed powers, such as k=5: C(10,5)·2^5·3^5 = 252·32·243 = 1,959,552.
Negative inputs flip signs predictably. If a is negative, terms alternate by k parity because a^k changes sign when k is odd. If b is negative, signs alternate by n−k parity. These patterns are useful for checking work: a missing minus sign usually breaks the expected alternation immediately.
This calculator uses string‑based big‑integer arithmetic so results remain exact when values overflow standard 64‑bit ranges. For example, C(100,50) is about 1.01×10^29, far beyond typical integer storage. Exact computation supports n up to 300 without rounding drift, making it suitable for proofs, assignments, and test banks.
The term table reports k, C(n,k), a^k, b^(n−k), and the final coefficient. That decomposition turns the expansion into a verifiable pipeline: first validate C(n,k), then validate powers, then confirm the product. Instructors can grade faster by spot‑checking a few k values rather than re‑expanding the whole expression.
The coefficient graph plots |coefficient| versus k. When a and b have similar magnitudes, the curve is often near‑symmetric around k≈n/2, with the largest values near the center. When |a|>|b|, peaks shift toward larger k; when |b|>|a|, peaks shift toward k close to 0. This visual cue helps learners connect algebraic structure to numerical behavior.
For teaching, ask students to predict where the maximum occurs before calculating. Then compare the predicted k with the plotted maximum and explain the difference using a and b. In real datasets.
It expands (a·x + b)n where a and b are integers and n is a non‑negative integer. The output is a full polynomial with exact integer coefficients.
Coefficients include C(n,k) and power factors. Middle binomial coefficients grow quickly as n increases, and multiplying by ak and bn−k can amplify them further.
It plots log10(|coefficient|) versus k, so you can see where magnitudes peak and whether the distribution shifts toward higher or lower powers based on |a| and |b|.
The calculator uses string‑based big‑integer arithmetic for addition, multiplication, and exact division while building C(n,k). This avoids floating rounding and integer overflow for large expansions.
Yes. Negative inputs change term signs predictably through ak and bn−k. The expansion and the table will reflect the correct alternation and final signed coefficients.
Exports include inputs (a, b, n, variable) and the full coefficient table: k, C(n,k), ak, bn−k, and the computed coefficient. The PDF also prints the expanded expression.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.