Binomial Theorem
The Binomial Theorem provides a formula for expanding any positive integer power of a binomial (a + b)ⁿ as a sum of terms of the form C(n,k) × aⁿ⁻ᵏ × bᵏ, where C(n,k) are binomial coefficients. It eliminates the need to multiply out the expression repeatedly and reveals the coefficient pattern known as Pascal's triangle. The theorem has applications in probability theory, combinatorics, calculus approximations, and algebraic identities.
Key Formula
(a + b)^n = Σ C(n,k) × a^(n-k) × b^k, for k = 0 to n
LaTeX: (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
| Symbol | Meaning | Unit |
|---|---|---|
| a, b | Terms of the binomial | dimensionless |
| n | Non-negative integer exponent | dimensionless |
| k | Index of summation (0 to n) | dimensionless |
| C(n,k) | Binomial coefficient n! / (k!(n−k)!) | dimensionless |
Worked Example
Problem
Expand (x + 2)⁴ using the Binomial Theorem.
Solution
Step 1: Apply (a+b)⁴ = C(4,0)a⁴ + C(4,1)a³b + C(4,2)a²b² + C(4,3)ab³ + C(4,4)b⁴. Step 2: Substitute a = x, b = 2: = 1·x⁴ + 4·x³·2 + 6·x²·4 + 4·x·8 + 1·16 = x⁴ + 8x³ + 24x² + 32x + 16.
Answer
(x + 2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16.
Binomial Coefficients C(n,k) — Pascal's Triangle (n = 0 to 5)
| n | Row of Coefficients | Expansion Pattern |
|---|---|---|
| 0 | 1 | (a+b)⁰ = 1 |
| 1 | 1 1 | (a+b)¹ = a + b |
| 2 | 1 2 1 | (a+b)² = a²+2ab+b² |
| 3 | 1 3 3 1 | (a+b)³ = a³+3a²b+3ab²+b³ |
| 4 | 1 4 6 4 1 | (a+b)⁴ = a⁴+4a³b+6a²b²+4ab³+b⁴ |
| 5 | 1 5 10 10 5 1 | (a+b)⁵ = a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵ |
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Related Terms
Mathematical Series
A mathematical series is the sum of the terms of a sequence, either finite or infinite. For a finite series the sum is always a specific number, while an infinite series converges to a finite limit only if its terms decrease fast enough. Series are foundational in calculus, number theory, physics, and engineering — from computing π to modelling oscillations and signal processing.
Exponent
An exponent (also called a power or index) indicates how many times a base number is multiplied by itself. Written as aⁿ, where a is the base and n is the exponent, it represents repeated multiplication in a compact form. Exponents are essential in scientific notation, polynomial expressions, and exponential growth models.
Arithmetic Sequence
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers in which the difference between consecutive terms is constant, called the common difference d. The nth term is given by aₙ = a₁ + (n−1)d, where a₁ is the first term. Arithmetic sequences model uniform increments such as salary increases, regular savings, and equally spaced physical measurements.
From Latin "binomium" (two names/terms), coined from "bi-" (two) and "nomen" (name). The theorem was known to Chinese mathematician Yang Hui (1261) and Persian mathematician Al-Karaji (c. 1000 AD); Isaac Newton generalised it to fractional exponents in 1665.