Binomial Expansion

IB Mathematics Analysis and Approaches HL· 1.8 Binomial expansion· 20 min read

1. The Binomial Theorem and Coefficient Calculation★★☆☆☆⏱ 7 min

📘 Definition

Binomial Theorem

For positive integer $n$

The expansion of $(a + b)^{n}$ as a sum of terms of the form $(k n) a^{n - k} b^{k}$ for all $k$ from $0$ to $n$

Example:

For $n = 2$ : $(a + b)^{2} = a^{2} + 2 ab + b^{2}$

There are two common methods to calculate binomial coefficients: constructing Pascal's triangle, or calculating directly with the combination formula: $(k n) = \frac{n !}{k ! ( n - k )!}$ . Pascal's triangle is built by starting with a top row of 1, then each new entry is the sum of the two entries directly above it. The $n$ th row (starting counting from 0) gives coefficients for $(a + b)^{n}$ .

📐 Worked Example

Expand $(x + 2)^{3}$ using Pascal's triangle

1
For $n = 3$ , we use the 3rd row of Pascal's triangle, which is $[1, 3, 3, 1]$
2
Write all terms of the expansion:
3
$1 \cdot x^{3} \cdot 2^{0} + 3 \cdot x^{2} \cdot 2^{1} + 3 \cdot x^{1} \cdot 2^{2} + 1 \cdot x^{0} \cdot 2^{3}$
4
Simplify each term to get the full expansion:
5
$x^3 + 6x^2 + 12x + 8$

2. Finding a Specific Term★★★☆☆⏱ 8 min

Most exam questions do not require you to expand the entire binomial. Instead, they ask for the coefficient of a specific power of $x$ , or a specific term. We can find this directly using the general term formula, no full expansion needed.

📘 Definition

General Term of a Binomial Expansion

For $(a + b)^{n}$ , the $(k + 1)$ th term (starting counting from $k = 0$ ) is: $T_{k + 1} = (k n) a^{n - k} b^{k}$

Example:

In $(x + 3)^{5}$ , the 3rd term has $k = 2$ , so $T_{3} = 90 x^{3}$

📐 Worked Example

Find the coefficient of $x^{4}$ in the expansion of $(2 x - 1)^{6}$

1
We have $n = 6$ , $a = 2 x$ , $b = - 1$ . We need the power of $x$ to be 4, so $n - k = 4 ⟹ k = 2$
2
Substitute into the general term formula:
3
$T_{3} = \binom{6}{2} (2x)^{4} (-1)^2$
4
Calculate the coefficient step-by-step:
5
$\binom{6}{2} = 15, \quad (2)^4 = 16, \quad (-1)^2 = 1$
6
Multiply to get the final coefficient: $15 \times 16 \times 1 = 240$

Exam tip:

Always remember the first term uses $k = 0$ , so the $m$ th term uses $k = m - 1$ , not $k = m$

3. Binomial Expansion for Approximation★★★☆☆⏱ 5 min

Binomial expansion can be used to find approximate values of powers of numbers very close to 1, without using a calculator. This works because higher powers of small numbers become negligible very quickly, so we can ignore them.

📐 Worked Example

Use binomial expansion to approximate $1.0 2^{5}$ correct to 3 decimal places

1
Rewrite $1.0 2^{5}$ as $(1 + 0.02)^{5}$ and expand:
2
$(1 + 0.02)^5 = \binom{5}{0}1^5(0.02)^0 + \binom{5}{1}1^4(0.02)^1 + \binom{5}{2}1^3(0.02)^2 + \binom{5}{3}1^2(0.02)^3 + ...$
3
Calculate each term:
4
$= 1 + 0.1 + 0.004 + 0.00008 + ...$
5
All terms after the third are smaller than 0.0005, so they do not affect the 3rd decimal place. Rounding gives:
6
$1.02^5 \approx 1.104$

4. Common Pitfalls

Wrong move:

Counting terms starting from $k = 1$ instead of $k = 0$ , so using $k = m$ for the $m$ th term

Why:

The first term of the expansion has $k = 0$ , so term number is always one higher than the value of $k$

Correct move:

Always use $k = m - 1$ when finding the $m$ th term of an expansion

Wrong move:

Forgetting to raise the coefficient of the variable to the correct power, e.g. taking $(2 6)$ as the coefficient of $x^{4}$ in $(2 x - 1)^{6}$

Why:

The entire variable term (including its coefficient) is raised to the power $n - k$ , not just the variable itself

Correct move:

Always include the coefficient of $x$ when calculating the power term

Wrong move:

Ignoring the negative sign on the second term of the binomial

Why:

For $(a - b)^{n}$ , the second term $b$ is negative, so odd powers of $k$ will give negative terms

Correct move:

Always write $b = - c$ for $(a - c)^{n}$ and include the negative sign when raising to the power $k$

Wrong move:

Using the positive integer binomial formula for negative or fractional exponents

Why:

This formula is only valid for positive integer $n$ ; using it for other exponents will give incorrect results

Correct move:

The IB AI HL syllabus only requires binomial expansion for positive integer $n$ , so you will never need to use it for other exponent types

5. Quick Reference Cheatsheet

Concept	Formula / Rule
Binomial Theorem	$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k}$
General Term	$T_{k + 1} = (k n) a^{n - k} b^{k}$ , $k$ starts at 0
Binomial Coefficient	$(k n) = \frac{n !}{k ! ( n - k )!}$
Pascal's Triangle	Entry = sum of the two entries directly above it
Approximation	Ignore small higher powers of $b$ for $(1 + b)^{n}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find 4th term of binomial expansion
2022 · 1
Approximate value via binomial expansion
2023 · 2
Find coefficient of specific term

Going deeper

videoBinomial Theorem and Pascal's TriangleVisual introduction to coefficient patterns

What's Next

Binomial expansion is a foundational algebraic tool for IB Math AI HL, used across multiple topics including binomial probability distributions, polynomial modeling, and approximate numerical calculations. Mastering the skill of quickly finding specific terms and coefficients will save you significant time on multi-step exam questions that rely on this sub-topic as a prerequisite. This sub-topic only covers expansion for positive integer exponents, which aligns with all requirements for the AI HL syllabus. You can now apply this knowledge to related topics in combinatorics, probability, and polynomials.