Binomial Theorem Reference Cheat Sheet

Key Facts

• The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$ for any nonnegative integer $n$.
• The binomial coefficient is $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, where $0 \le k \le n$.
• The general term of $(a + b)^n$ is $T_{k+1}=\binom{n}{k}a^{n-k}b^k$ because the first term occurs when $k=0$.
• In the expansion of $(a + b)^n$, there are $n+1$ terms before combining any like terms.
• For each term in $(a + b)^n$, the exponents add to $n$, so $a^{n-k}b^k$ has total degree $n$.
• The coefficients in $(a + b)^n$ match row $n$ of Pascal's triangle when the top row is row $0$.
• For subtraction, $(a - b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}(-b)^k$, so signs alternate when $b$ is positive.
• The symmetry rule $\binom{n}{k}=\binom{n}{n-k}$ means matching coefficients from opposite ends of the expansion are equal.

Vocabulary

Binomial

A binomial is an algebraic expression with two terms, such as $a+b$ or $x-3$.

Binomial theorem

The binomial theorem is the formula $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k$ for expanding a binomial raised to a nonnegative integer power.

Binomial coefficient

A binomial coefficient $\binom{n}{k}$ is the number that multiplies a term in a binomial expansion and equals $\frac{n!}{k!(n-k)!}$.

Factorial

A factorial $n!$ is the product of all positive integers from $1$ to $n$, with $0! = 1$.

General term

The general term is a formula such as $T_{k+1}=\binom{n}{k}a^{n-k}b^k$ that describes any term in an expansion.

Pascal's triangle

Pascal's triangle is a triangular arrangement of numbers where each interior number is the sum of the two numbers above it.