Math101 · Independent Ontario learning library
Math101Binomial Theorem
The binomial theorem expands a nonnegative integer power of a sum: $(a+b)^n=\sum_{k=0}^n\binom nk a^{n-k}b^k$.
The theorem expands large powers efficiently and connects algebra with combinations, probability, calculus, and approximation.
Intuition and core definition
The binomial theorem expands a nonnegative integer power of a sum: $(a+b)^n=\sum_{k=0}^n\binom nk a^{n-k}b^k$. The coefficients count ways to choose which $k$ of the $n$ factors contribute $b$. Exponents on $a$ decrease while exponents on $b$ increase, and every term has total degree $n$.
Notation, language, and conditions
$\binom nk=\frac{n!}{k!(n-k)!}$ is read “$n$ choose $k$.” The expansion has $n+1$ terms before like-term coincidences. For $(a-b)^n$, treat the second term as $-b$, producing alternating signs according to $(-b)^k$.
Why this idea matters
The binomial theorem organizes every term of a power using combinatorial coefficients, avoiding repeated distribution while preserving exponent patterns.
A dependable method
- Identify the two binomial terms, including any coefficient or negative sign, and the integer exponent $n$.
- List coefficients from Pascal’s triangle or compute $\binom nk$.
- Write powers of the first term from $n$ down to $0$ and powers of the second from $0$ up to $n$.
- Apply coefficients and simplify signs and numerical powers.
- Check term count, total degree, endpoints, and a simple substitution.
