# Binomial expansion and binomial theorem

> IB Mathematics Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u1-binomial-expansion-and-binomial-theorem/

This subtopic covers the binomial theorem for positive integer exponents, calculation of binomial coefficients, full expansion of binomials, and finding specific terms without full expansion — a frequent low-to-medium marks exam topic.

**Prerequisites:** [Factorial notation and combinations](https://www.owlsprep.com/study/ib-math-aa-sl-u1-combinations-and-permutations/); Basic algebraic exponent rules

## Learning objectives

- State the binomial theorem for positive integer exponents
- Calculate binomial coefficients using factorials and Pascal's triangle
- Expand binomial expressions of any order up to n=10
- Find a specific term in a binomial expansion without full expansion

## Binomial Coefficients

**Binomial Coefficient** — A value representing the number of ways to choose k items from n total items, used to find the coefficient of each term in a binomial expansion.

*Notation:* \binom{n}{k} = \frac{n!}{k!(n-k)!}

*Example:* \binom{5}{2} = 10

Binomial coefficients can also be found using Pascal's Triangle, where the nth row (starting counting from row 0) gives all coefficients for $(a+b)^n$. The first 5 rows are shown below:

$$\begin{aligned} \text{Row } 0: & \quad 1 \\ \text{Row } 1: & \quad 1 \quad 1 \\ \text{Row } 2: & \quad 1 \quad 2 \quad 1 \\ \text{Row } 3: & \quad 1 \quad 3 \quad 3 \quad 1 \\ \text{Row } 4: & \quad 1 \quad 4 \quad 6 \quad 4 \quad 1 \end{aligned}$$

**Worked example:** Calculate $\binom{6}{3}$ using the factorial formula, then confirm with Pascal's triangle.

1. Substitute n=6, k=3 into the factorial formula:
2. $$\binom{6}{3} = \frac{6!}{3! (6-3)!} = \frac{6 \times 5 \times 4 \times 3!}{3! \times 3 \times 2 \times 1}$$
3. Cancel $3!$ from numerator and denominator, then simplify:
4. $$\frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20$$
5. Row 6 of Pascal's triangle is $[1, 6, 15, 20, 15, 6, 1]$. The 4th entry (index starting at 0) is 20, which matches the calculation.

> **Exam tip:** For n ≤ 6, Pascal's triangle is faster than factorial calculation and less prone to arithmetic error.

## The Binomial Theorem and Full Expansion

**Binomial Theorem** — A general formula to expand any power of a binomial sum $(a+b)^n$ for positive integer n.

*Notation:* (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k

> **warning**
>
> If your binomial has a negative term such as $(a - b)$, rewrite it as $(a + (-b))$ to avoid sign errors. All odd powers of $-b$ will be negative.

**Worked example:** Expand $(2x - 3)^3$ fully using the binomial theorem.

1. Align the binomial with the formula: $a=2x$, $b=-3$, $n=3$.
2. Write out each term using the binomial sum:
3. $$\binom{3}{0}(2x)^3(-3)^0 + \binom{3}{1}(2x)^2(-3)^1 + \binom{3}{2}(2x)^1(-3)^2 + \binom{3}{3}(2x)^0(-3)^3$$
4. Calculate each term individually:
5. 1. First term: $1 \times 8x^3 \times 1 = 8x^3$
2. Second term: $3 \times 4x^2 \times (-3) = -36x^2$
3. Third term: $3 \times 2x \times 9 = 54x$
4. Fourth term: $1 \times 1 \times (-27) = -27$
6. Combine all terms for the final expansion:
7. $$(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27$$

## Finding a Specific Term

A very common IB exam question asks for the coefficient of a specific term (or the constant term) in a binomial expansion, without requiring full expansion. To do this, you use the general term of the expansion to solve for the k value that matches your required exponent.

For an expansion of $(ax^m + b)^n$, the general (k+1)th term (starting from k=0) is:

$$T_{k+1} = \binom{n}{k} (ax^m)^{n-k} b^k$$

**Worked example:** Find the constant term in the expansion of $\left(x^2 + \frac{2}{x}\right)^6$.

1. Identify values: $a=1$, $m=2$, $b=2$, $n=6$. Substitute into the general term:
2. $$T_{k+1} = \binom{6}{k} (x^2)^{6-k} \left(\frac{2}{x}\right)^k$$
3. Simplify the exponent of $x$:
4. $$\text{Exponent of } x = 2(6-k) - k = 12 - 3k$$
5. A constant term has an exponent of 0, so solve for k:
6. $$12 - 3k = 0 \implies k = 4$$
7. Substitute k=4 back to find the constant term:
8. $$T_{5} = \binom{6}{4} (x^2)^{2} \left(\frac{2}{x}\right)^4 = 15 \times x^4 \times \frac{16}{x^4} = 240$$
9. The constant term is 240.

> **Exam tip:** Remember that k starts at 0, so the mth term uses k = m-1, not k=m.

## Common pitfalls

- **Wrong:** Forgetting negative signs when expanding $(a - b)^n$
  - Why it fails: Treating the second term as positive b instead of -b, leading to wrong signs on odd-powered terms
  - Correct: Rewrite $(a - b)^n$ as $(a + (-b))^n$ before substitution, to explicitly track negative signs
- **Wrong:** Using k equal to the term number (e.g. k=5 for the 5th term)
  - Why it fails: The first term of the expansion is k=0, so the k value is offset by 1
  - Correct: For the mth term, use k = m-1 when calculating the general term
- **Wrong:** Swapping exponents: writing $\binom{n}{k} a^k b^{n-k}$ instead of $\binom{n}{k} a^{n-k} b^k$
  - Why it fails: Misaligning a and b with the exponents in the formula
  - Correct: Clearly label which term in your binomial is a and which is b before applying the formula
- **Wrong:** Forgetting to raise the constant coefficient of a term to the correct power
  - Why it fails: Only raising the variable part to the exponent, ignoring the constant coefficient
  - Correct: Remember $(cx^m)^p = c^p x^{mp}$, so always calculate the coefficient's power when expanding

## Cheatsheet

| Concept | Formula / Rule |
| --- | --- |
| Binomial Coefficient | $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ |
| Binomial Theorem | $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$ |
| General Term | $T_{k+1} = \binom{n}{k} (ax^m)^{n-k} b^k$ |
| Constant Term Condition | Set exponent of $x = 0$, solve for $k$ |
| Pascal's Triangle Entry | Entry = sum of two entries directly above |

## What's next

Binomial expansion is a foundational algebraic skill for IB AA SL that reappears across multiple topics, most notably in probability with the binomial distribution. The combinatorial reasoning behind binomial coefficients also underpins many counting problems for probability questions, which regularly appear on both Paper 1 and Paper 2. Even beyond exam questions, understanding the structure of binomial expansions makes expanding higher-order polynomials much faster than repeated distribution, and helps with finding derivatives or integrals of polynomial expressions. Mastery of this subtopic will also prepare you for further work on sequences and series in later units.

- [Functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-overview/)
- [Function concepts, domain, range and notation](https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-concepts-domain-range-and/)

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