Binomial Theorem
Calculus & Advanced MathThe Binomial Theorem gives a formula for expanding (a + b)^n as a sum of terms involving binomial coefficients.
Formula
(a+b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k
Definition
The Binomial Theorem is a shortcut for expanding expressions like (a + b)^n. Instead of multiplying (a + b) by itself n times, you use a formula with coefficients from Pascal's Triangle.
Example
$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$. The coefficients $1, 3, 3, 1$ come from the $4$th row of Pascal's Triangle.
Key Insight
Without the Binomial Theorem, expanding $(a + b)^{10}$ would require multiplying $10$ times. With it, you write out $11$ terms instantly.
Definition
$(a+b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k$, where $C(n,k) = n!/(k!(n-k)!)$ is the binomial coefficient, giving the number of ways to choose $k$ items from $n$. The sum of all coefficients is $2^n$ (set $a = b = 1$).
Example
$(2x - 3)^4$: $a=2x$, $b=-3$. Term $k=2$: $C(4,2)(2x)^2(-3)^2 = 6 \times 4x^2 \times 9 = 216x^2$. Full expansion: $16x^4 - 96x^3 + 216x^2 - 216x + 81$.
Key Insight
The middle term of $(a + b)^n$ (when $n$ is even) has the largest coefficient, reflecting the maximum of the binomial distribution at its mean.
Definition
The Binomial Theorem can be proved by induction or combinatorially: $C(n,k)$ counts the number of ways to choose $k$ copies of $b$ from $n$ factors of $(a+b)$. Newton's generalized binomial theorem extends to non-integer and even complex exponents: $(1+x)^\alpha = \sum_{k=0}^{\infty} C(\alpha,k) x^k$ for $|x| < 1$.
Example
Newton's binomial: $(1+x)^{1/2} = 1 + x/2 - x^2/8 + x^3/16 - \ldots$ Used to approximate $\sqrt{1.04}$: set $x=0.04$, get approximately $1 + 0.02 - 0.0002 = 1.0198$. Actual: $1.0198039\ldots$
Key Insight
Newton's generalized binomial series is a power series convergent for $|x| < 1$. It is a precursor to Taylor series and reveals how binomial coefficients extend naturally to the complex numbers via the Gamma function.