Binomial Theorem (Positive Integer n)

CIE IGCSE Additional Mathematics· 12.1, 12.2· 25 min read

1. Pascal's Triangle and Basic Binomial Expansion★★☆☆☆⏱ 10 min

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📘 Definition

Binomial Theorem (Positive Integer n)

For any positive integer $n$ , $(a + b)^{n} = a^{n} + (1 n) a^{n - 1} b + (2 n) a^{n - 2} b^{2} + ... + b^{n}$ , where coefficients correspond to entries in the $n$ -th row of Pascal's triangle.

Example:

$(x + 2)^{3} = x^{3} + 3 x^{2} (2) + 3 x (2)^{2} + 2^{3} = x^{3} + 6 x^{2} + 12 x + 8$

Pascal's triangle starts with a row for $n = 0$ (value [1]), and each subsequent row is formed by adding adjacent entries from the row above. Each entry in row $n$ is the coefficient of the corresponding term in the expansion of $(a + b)^{n}$ .

📐 Worked Example

Expand $(2 x - 3)^{4}$ using Pascal's triangle.

1
1. Identify row 4 of Pascal's triangle: $[1, 4, 6, 4, 1]$
2
1. Substitute $a = 2 x$ , $b = - 3$ , $n = 4$ into the expansion formula:
3
$(2x)^4 + 4(2x)^3(-3) + 6(2x)^2(-3)^2 + 4(2x)(-3)^3 + (-3)^4$
4
1. Simplify each term individually:
5
$16x^4 + 4(8x^3)(-3) + 6(4x^2)(9) + 4(2x)(-27) + 81$
6
1. Combine simplified terms to get the final expansion:
7
$16x^4 - 96x^3 + 216x^2 - 216x + 81$

Exam tip:

Always include the sign of the second term when substituting into the expansion, especially if the term is negative, to avoid sign errors.

2. nCr Notation and General Term Formula★★★☆☆⏱ 12 min

✓ Calculator OK

📘 Definition

nCr (Binomial Coefficient)

\binom{n}{r}

The number of ways to choose $r$ items from $n$ distinct items, calculated as $(r n) = \frac{n !}{r ! ( n - r )!}$ , where $n! = n \times (n - 1) \times ... \times 1$ .

Example:

$(2 5) = \frac{5 !}{2 ! 3 !} = 10$

The general term of a binomial expansion allows you to find a specific term without expanding the entire expression. The $(r + 1)$ -th term of $(a + b)^{n}$ is given by:

T_{r+1} = \binom{n}{r} a^{n-r} b^r

📐 Worked Example

Find the coefficient of $x^{5}$ in the expansion of $(3 x + 1)^{8}$ .

1
1. Identify $n = 8$ , $a = 3 x$ , $b = 1$ . We need the term where the power of $x$ is 5, so $n - r = 5$ → $r = 3$ .
2
1. Substitute into the general term formula for $r = 3$ :
3
$T_{4} = \binom{8}{3} (3x)^{5} (1)^3$
4
1. Calculate the coefficient components: $(3 8) = 56$ , $3^{5} = 243$
5
1. Multiply to get the final coefficient: $56 \times 243 = 13608$

Exam tip:

Note that $r$ starts at 0, so the $r$ -th combination corresponds to the $(r + 1)$ -th term. Mixing up this offset is one of the most common exam errors for this topic.

3. Calculating the Term Independent of x★★★★☆⏱ 10 min

📘 Definition

Term Independent of x

The constant term in a binomial expansion where the total exponent of $x$ is 0, so it contains no $x$ variable, only a numerical value.

To find the term independent of $x$ , rewrite all terms with $x$ using positive or negative exponents, set the total exponent of $x$ in the general term equal to 0, solve for $r$ , and substitute back into the general term if $r$ is a valid non-negative integer between 0 and $n$ .

📐 Worked Example

Find the term independent of $x$ in the expansion of $(2 x + \frac{1}{x ^{2}})^{9}$ .

1
1. Write the general term for the expansion:
2
$T_{r+1} = \binom{9}{r} (2x)^{9 - r} (x^{-2})^r$
3
1. Simplify the total exponent of $x$ : $(9 - r) - 2 r = 9 - 3 r$
4
1. Set exponent equal to 0 and solve for $r$ : $9 - 3 r = 0$ → $r = 3$
5
1. Substitute $r = 3$ back into the general term:
6
$\binom{9}{3}(2x)^{6}(x^{-2})^{3} = \binom{9}{3}\,2^{6}\,x^{6}\cdot x^{-6} = 84 \times 64 = 5376$

Exam tip:

Rewrite fractions with $x$ in the denominator using negative exponents first to avoid mistakes when summing exponents of $x$ .

4. Exam-Style Binomial Application Problems★★★★☆⏱ 15 min

Exam questions often ask you to use binomial expansions for approximations of decimal values, or to find coefficients in products of binomial expressions. For approximation questions, you only need to calculate terms that affect the required number of decimal places.

📐 Worked Example

Use the binomial theorem to find the value of $(1.02)^{5}$ correct to 3 decimal places.

1
1. Rewrite $1.02$ as $1 + 0.02$ , so $n = 5$ , $a = 1$ , $b = 0.02$
2
1. Expand up to terms that affect 3 decimal places:
3
$(1 + 0.02)^5 = 1 + \binom{5}{1}(0.02) + \binom{5}{2}(0.02)^2 + \binom{5}{3}(0.02)^3 + ...$
4
1. Calculate each non-negligible term:
5
$1 + 5(0.02) + 10(0.0004) + 10(0.000008) = 1 + 0.1 + 0.004 + 0.00008 = 1.10408$
6
1. Round to 3 decimal places: $1.104$

Exam tip:

For Paper 1 non-calculator questions, use Pascal's triangle for small values of $n$ to save time instead of calculating nCr manually.

5. Common Pitfalls

Wrong move:

Using $r$ instead of $r + 1$ for the term number

Why:

The general term $T_{r + 1}$ uses $r$ starting at 0, so $r = 0$ corresponds to term 1, not term 0.

Correct move:

Always add 1 to $r$ to get the term position, or solve for $r$ first by matching the required exponent of $x$ .

Wrong move:

Forgetting the negative sign of the second term when expanding $(a - b)^{n}$

Why:

The negative sign is part of $b$ , so it is raised to the power of $r$ , leading to alternating signs for negative $b$ .

Correct move:

Write $b$ explicitly as a negative value before substituting into the expansion formula to avoid sign errors.

Wrong move:

Using fractional or negative $n$ for binomial expansions

Why:

This is A Level content not tested in 0606, and will produce incorrect results for positive integer $n$ questions.

Correct move:

Only apply the binomial theorem for positive integer $n$ as specified in the question.

Wrong move:

Miscalculating the exponent of $x$ when finding the term independent of $x$

Why:

Terms with $x$ in the denominator have negative exponents, which are often added incorrectly if not rewritten first.

Correct move:

Rewrite all $x$ terms with negative exponents first, then sum exponents to set equal to 0.

Wrong move:

Generating the wrong row of Pascal's triangle for $n$

Why:

Pascal's triangle starts at $n = 0$ (row 0 = [1]), so row $n$ corresponds to exponent $n$ , not the row number if counting from 1.

Correct move:

Count rows starting from 0, so for $n = 4$ , use the row with 5 entries starting and ending with 1.

6. Quick Reference Cheatsheet

Concept	Formula/Rule	Exam Note
Basic Expansion	$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$	n only positive integer
Pascal's Triangle	Row n entry r = $(r n)$	Row 0 = [1], row 1 = [1,1]
General Term	$T_{r + 1} = (r n) a^{n - r} b^{r}$	Term number = r+1, r starts at 0
Term Independent of x	Set total x exponent = 0, solve for r	Only valid if r is integer 0 ≤ r ≤ n
nCr Calculation	$(r n) = \frac{n !}{r ! ( n - r )!} = (n - r n)$	Use calculator nCr function for Paper 2

7. Frequently Asked

Do I need to memorize Pascal's triangle for 0606?

You can either generate Pascal's triangle up to the required n or use the nCr formula; both are acceptable. The nCr function on your calculator is faster for larger n in Paper 2.

Can I use fractional or negative n for binomial expansions in 0606?

No, only positive integer n is tested in 0606. Fractional/negative index expansions are A Level content and not required for this exam.

Why is the general term written as T_{r+1} instead of T_r?

r starts at 0, so the first term corresponds to r=0 (T₁), second to r=1 (T₂), etc. This offset aligns with the nCr index for the binomial coefficient.

Going deeper

What's Next

Now that you have mastered the binomial theorem for positive integer n, you can apply this skill to solve more complex series problems, including combinations with calculus and algebraic simplification questions that frequently appear in CIE IGCSE Additional Mathematics 0606 papers. You are now ready to practice past paper questions on this topic, and move on to other core series topics such as arithmetic and geometric progressions. Make sure to practice both calculator and non-calculator questions to prepare for both Paper 1 and Paper 2.