11.6 Binomial Theorem

Binomial Theorem and Expansions

The binomial theorem gives you a formula for expanding expressions like $(x + y)^n$ without multiplying the binomial by itself over and over. It connects algebra with combinatorics through binomial coefficients, and it shows up constantly in probability, series work, and later calculus topics.

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Binomial theorem application

The theorem expands any expression of the form $(x + y)^n$, where $n$ is a non-negative integer, into a sum of $n + 1$ terms.

The general formula:

$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

Here's what each piece means:

• $\sum_{k=0}^{n}$ tells you to add up all the terms as $k$ goes from 0 to $n$
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$
• $n!$ (read "n factorial") means $n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1$, with $0! = 1$ by definition

Each individual term follows a predictable pattern:

• The exponent on $x$ starts at $n$ and decreases by 1 each term
• The exponent on $y$ starts at 0 and increases by 1 each term
• The exponents on $x$ and $y$ always add up to $n$

For example, $(a + b)^3 = \binom{3}{0}a^3 + \binom{3}{1}a^2b + \binom{3}{2}ab^2 + \binom{3}{3}b^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

This works for any binomial raised to a power, including expressions like $(2x + 3)^4$ or $(a - b)^6$. When a term is negative, like $(a - b)^6$, treat it as $(a + (-b))^6$ so the signs alternate correctly.

Specific term calculation

You can find any single term in a binomial expansion without expanding the whole thing. This saves a ton of time when $n$ is large.

Steps to find a specific term:

1. Identify which term you need. The $(k+1)$th term corresponds to the value $k$ in the formula. So the 1st term uses $k = 0$, the 2nd term uses $k = 1$, the 4th term uses $k = 3$, and so on.
2. Calculate the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
3. Build the term: $\binom{n}{k} \cdot x^{n-k} \cdot y^k$.
4. Simplify by evaluating any powers and multiplying everything together.

Example: Find the 4th term in the expansion of $(3x - 2y)^6$.

1. The 4th term means $k = 3$.

2. Compute the coefficient: $\binom{6}{3} = \frac{6!}{3! \cdot 3!} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20$

3. Build the term: $20 \cdot (3x)^{6-3} \cdot (-2y)^3$

4. Simplify each piece:

   • $(3x)^3 = 27x^3$
   • $(-2y)^3 = -8y^3$ (odd exponent keeps the negative)
   • $20 \cdot 27 \cdot (-8) = -4320$

5. Result: $-4320x^3y^3$

Watch the signs carefully. When the binomial has a subtraction, the terms with odd powers of $y$ will be negative.

Pascal's triangle interpretation

Pascal's Triangle is a triangular array where each entry is the sum of the two entries directly above it. The first few rows look like this:

Every row starts and ends with 1. Rows are numbered starting from 0.

The number in row $n$, position $k$ (also starting from 0) equals the binomial coefficient $\binom{n}{k}$. So the entry in row 4, position 2 is $\binom{4}{2} = 6$, which you can verify from the table above.

This means Pascal's Triangle gives you the coefficients for any binomial expansion at a glance. For $(x+y)^4$, just read row 4: the coefficients are 1, 4, 6, 4, 1.

Key properties:

• The sum of all entries in row $n$ equals $2^n$ (for example, row 3: $1 + 3 + 3 + 1 = 8 = 2^3$)
• The triangle is symmetric: $\binom{n}{k} = \binom{n}{n-k}$
• Each entry represents a combination: $\binom{n}{k}$ counts the number of ways to choose $k$ items from $n$ items when order doesn't matter

For small values of $n$, reading coefficients off Pascal's Triangle is faster than computing factorials. For larger values, the formula $\frac{n!}{k!(n-k)!}$ becomes more practical.

Additional Concepts in Binomial Expansion

• Polynomial expansion: The binomial theorem is a specific case of expanding polynomials. Multinomial expansion generalizes the idea to expressions with more than two terms, like $(x + y + z)^n$.
• Combinations vs. permutations: Binomial coefficients count combinations (order doesn't matter). Permutations, where order does matter, use the formula $\frac{n!}{(n-k)!}$ and come up in more complex counting and probability problems.
• Connection to probability: Binomial coefficients appear directly in the binomial probability distribution, which models the number of successes in $n$ independent trials. You'll see this in statistics and in AP-level probability work.