Binomial Expansion (a + bx)ⁿ — Coefficients & Terms
Enter (a + bx)ⁿ to expand it fully, list each coefficient, and get the coefficient of the x^k term.
Input Data
Results
At a glance:Binomial theorem: (a + bx)ⁿ = Σ_{k=0}^{n} C(n,k)·a^{n−k}·(bx)^k, with C(n,k)=n!/[k!(n−k)!]. The x^k-term coefficient is C(n,k)·a^{n−k}·b^k; there are n+1 terms.
Formula
(a + bx)ⁿ = Σ_{k=0}^{n} C(n,k) a^{n−k} b^k x^k.
x^k-term coefficient = C(n,k)·a^{n−k}·b^k.
C(n,k) = n! / (k!(n−k)!).
$$(a+bx)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} x^{k}$$$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$How to Use
- Enter constant a, coefficient b and power n (non-negative integer).
- Enter the term index k you want (0 ≤ k ≤ n).
- The tool outputs the full expansion, all coefficients, and the x^k-term coefficient with its full term.
- Compare with the case studies to check your expansion.
Small binomial coefficients ((1+x)ⁿ)
| n | Coefficients C(n,k) | Terms |
|---|---|---|
| n=2 | 1, 2, 1 | 3 |
| n=3 | 1, 3, 3, 1 | 4 |
| n=4 | 1, 4, 6, 4, 1 | 5 |
| n=5 | 1, 5, 10, 10, 5, 1 | 6 |
| n=6 | 1, 6, 15, 20, 15, 6, 1 | 7 |
Coefficients are symmetric: C(n,k)=C(n,n−k); sum Σ C(n,k)=2ⁿ.
Case Studies
(1 + x)⁴
Expansion: 1 + 4x + 6x² + 4x³ + x⁴.
Coefficients are row 5 of Pascal's triangle: 1, 4, 6, 4, 1.
x²-term coefficient = C(4,2)·1²·1² = 6.
(2 + x)³
= 8 + 12x + 6x² + x³.
k=1 coefficient = C(3,1)·2²·1 = 3·4 = 12.
(1 + 2x)⁵
= 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵.
x³-term coefficient = C(5,3)·1²·2³ = 10·8 = 80.
Middle term of (x + 1)ⁿ
For even n the largest coefficient is at the middle k=n/2.
e.g. (1+x)⁶: x³ coefficient = C(6,3) = 20 is the maximum.
(a − bx)ⁿ with negative sign
Set b negative, e.g. (1−x)³ = 1 − 3x + 3x² − x³.
Odd-k terms are negative because b^k < 0.
Binomial coefficients in probability
C(n,k) is also the combination count in the binomial distribution.
e.g. exactly 2 heads in 4 coin flips: C(4,2)=6 ways.
FAQ
Can n be negative or fractional?
This tool requires n to be a non-negative integer (finite expansion). Negative or fractional powers give infinite series needing the generalized binomial theorem, outside this tool.
x^k term vs the 'k-th term'?
This tool labels by the power k of x (starting at 0). Conventionally the 'r-th term' starts at 1, i.e. k=r−1; mind the offset.
Negative coefficients — is that right?
If b is negative (e.g. (a−bx)ⁿ), odd-k terms have b^k < 0, so negative coefficients are correct.
Where is the largest coefficient?
For (1+x)ⁿ coefficients rise then fall, symmetric; even n peaks at the middle k=n/2, odd n has two equal middle peaks.
Relation to probability?
C(n,k) is exactly the number of ways to get k successes in n trials, and the coefficient of the binomial distribution — the same source.
Always n+1 terms?
Yes, a binomial with non-negative integer n has exactly n+1 terms (k=0 to n).
What if a or b is 0?
a=0 leaves only (bx)ⁿ; b=0 leaves only aⁿ. The tool still outputs correctly (other coefficients 0).
Overflow for large coefficients?
n is capped at 30; C(30,15)≈1.55×10⁸ is safe. For larger powers use a scientific calculator.
Related Tools
References
Content review: Calculatorism Science Team. Binomial theorem, C(n,k) and the coefficient formula verified. Results are for reference only.