Binomial Theorem

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

1. Identify the two binomial terms, including any coefficient or negative sign, and the integer exponent $n$.
2. List coefficients from Pascal’s triangle or compute $\binom nk$.
3. Write powers of the first term from $n$ down to $0$ and powers of the second from $0$ up to $n$.
4. Apply coefficients and simplify signs and numerical powers.
5. Check term count, total degree, endpoints, and a simple substitution.

Worked example

Representations and interpretation

Pascal’s triangle supplies coefficient rows, while a combinatorial selection diagram explains them: choosing $k$ copies of the second term among $n$ identical factors creates $\binom nk$ equal products.

Reasoning about variations

The theorem is not simply “raise each term”: $(a+b)^n\ne a^n+b^n$ for $n>1$ in general. For a requested single term, the general term $T_{k+1}=\binom nk a^{n-k}b^k$ avoids expanding everything.

Common mistakes

How to check your work

• Set variables to $1$ and compare the expansion with the original binomial value.
• Verify first and last terms are $a^n$ and $b^n$ including internal coefficients.
• Check each unsimplified term has exponent total $n$.

Practice

1. Find the coefficient of $x^2$ in $(x+2)^4$.
2. Expand $(a-b)^3$.
3. How many terms are in the standard expansion of $(p+q)^7$?

Answers and brief solutions

Synthesis and transfer

To find one coefficient in a high power, select the corresponding combination and complementary exponents; checking the total degree catches an incorrectly indexed term.

In $(x+2)^6$, the $x^4$ term occurs when two of the six factors contribute $2$, so its coefficient is $\binom62 2^2=60$. The exponents of $x$ and $2$ always add to six, and consecutive binomial coefficients follow Pascal's triangle. A single-term calculation therefore needs neither the entire expansion nor repeated distribution. Replacing the plus by a minus alternates signs according to the selected power, while changing the constants affects powers but not the combinatorial counts. Summing the coefficients by substituting $x=1$ gives a useful whole-expansion check: the result must equal $3^6$.

Related topics

• Binomial
• Combinations
• Special Products

Teaching and accessibility note