# The binomial series (1 + x)^n

> Edexcel International GCSE Further Pure Mathematics · 4PM1 (2016 spec)
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s6-the-binomial-series-n/

This guide teaches you to apply the binomial series for $(1 + x)^n$ for both positive integer exponents (finite expansions) and rational exponents (infinite, valid when $|x| < 1$). You will also learn to expand $(a + bx)^n$ and solve exam-style problems for coefficients and approximations.

**Prerequisites:** [Core GCSE binomial expansion for positive integer exponents](https://www.owlsprep.com/study/edexcel-gcse-maths-algebra-binomial-expansion/)

## Learning objectives

- Expand (1 + x)^n for positive integer n (terminating, no validity restriction)
- Expand (1 + x)^n for rational n, stating the validity condition |x| < 1
- Rewrite (a + bx)^n to apply the binomial series by factoring out the constant a
- Calculate specific terms, coefficients and approximate values using binomial expansions

## 1. Finite Expansion: n is a Positive Integer

When n is a positive integer, the binomial series for $(1 + x)^n$ produces a finite, terminating expansion with exactly n+1 terms. There is no validity restriction on x for this case, as the series stops after the $x^n$ term.

**Terminating binomial expansion** — A finite expansion of $(1 + x)^n$ for positive integer n, with no infinite terms, valid for all real values of x.

**Worked example:** Fully expand $(1 + 2x)^4$.

1. Recall the binomial series for positive integer n, matching the formula sheet:
2. $$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ... + x^n$$
3. Substitute $n = 4$ and $x = 2x$ into the formula:
4. $$1 + 4(2x) + \frac{4 \times 3}{2!}(2x)^2 + \frac{4 \times 3 \times 2}{3!}(2x)^3 + \frac{4!}{4!}(2x)^4$$
5. Simplify each term to get the final expansion:
6. $$1 + 8x + 24x^2 + 32x^3 + 16x^4$$

> **Exam tip:** For positive integer n, you can use either combination notation $\binom{n}{r}$ or the general term formula from the formula sheet, both produce identical results.

*Calculator:* allowed

## 2. Infinite Expansion: n is Rational, |x| < 1

When n is a negative integer, fraction, or any rational number that is not a positive integer, the binomial series produces an infinite expansion. This expansion is only mathematically valid when $|x| < 1$: you **must** state this condition in every exam answer for this case.

**Validity condition (rational n)** — The infinite expansion of $(1 + x)^n$ for $n \in \mathbb{Q}$ (non-positive integer) is only convergent when the absolute value of x is less than 1, written $|x| < 1$.

**Worked example:** Find the first three non-zero terms of the expansion of $(1 - x)^{-1/2}$, and state the range of x for which the expansion is valid.

1. Rewrite the expression as $(1 + (-x))^n$ where $n = -1/2$.
2. Apply the binomial series formula from the formula sheet:
3. $$1 + nx + \frac{n(n-1)}{2!}x^2 + ...$$
4. Substitute $n = -1/2$ and replace x with $-x$:
5. $$1 + (-1/2)(-x) + \frac{(-1/2)(-1/2 - 1)}{2!}(-x)^2 + ...$$
6. Simplify each term:
7. $$1 + \frac{1}{2}x + \frac{3}{8}x^2 + ...$$
8. State the validity condition: $|-x| < 1 \implies |x| < 1$

> **Exam tip:** Always substitute the entire argument of the bracket (e.g., $-x$ for $(1 - x)^n$) into the standard formula, not just the coefficient of x.

*Calculator:* allowed

## 3. Expanding $(a + bx)^n$ by Factoring Out a

To apply the standard $(1 + x)^n$ series to expressions of the form $(a + bx)^n$, first factor out the constant a from the bracket to rewrite the expression in the standard form. Adjust the validity condition to match the new argument of the $(1 + kx)$ term.

**Worked example:** Find the first three terms of the expansion of $(4 + x)^{-1}$, and state the validity condition.

1. Factor out 4 from the bracket, applying the exponent -1 to the factored constant:
2. $$(4(1 + x/4))^{-1} = 4^{-1} \left(1 + \frac{x}{4}\right)^{-1} = \frac{1}{4}\left(1 + \frac{x}{4}\right)^{-1}$$
3. Expand $\left(1 + \frac{x}{4}\right)^{-1}$ using the binomial series:
4. $$1 + (-1)\left(\frac{x}{4}\right) + \frac{(-1)(-2)}{2!}\left(\frac{x}{4}\right)^2 + ... = 1 - \frac{x}{4} + \frac{x^2}{16} + ...$$
5. Multiply through by $1/4$ to get the final expansion:
6. $$\frac{1}{4} - \frac{x}{16} + \frac{x^2}{64} + ...$$
7. State the validity condition: $\left|\frac{x}{4}\right| < 1 \implies |x| < 4$

> **Exam tip:** Never forget to apply the exponent n to the factored out constant term: $(a \times f(x))^n = a^n [f(x)]^n$, not $a [f(x)]^n$ — this is one of the most common mark-losing errors for this topic.

*Calculator:* allowed

## 4. Applications: Terms, Coefficients and Approximations

Exam questions frequently ask you to find specific terms (e.g., the coefficient of $x^3$) or use the first 2-3 terms of an expansion to approximate values of expressions for small x, where the higher-order terms are negligible.

**Worked example:** Use the first three terms of the expansion of $\sqrt{1 + 0.04}$ to find an approximate value of $\sqrt{1.04}$, giving your answer to 3 decimal places.

1. Rewrite the expression as $(1 + 0.04)^{1/2}$.
2. Calculate the first three terms of the expansion:
3. $$1 + \frac{1}{2}(0.04) + \frac{(1/2)(-1/2)}{2!}(0.04)^2$$
4. Simplify each term:
5. $$1 + 0.02 - 0.0002 = 1.0198$$
6. Round to 3 decimal places to get the final approximate value: 1.020

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to state the validity condition $|x| < 1$ for rational n expansions
  - Why it fails: Examiners explicitly award marks for stating this condition, so omitting it loses easy, guaranteed marks
  - Correct: Always write the validity condition immediately after any infinite binomial expansion answer, adjusting for the argument of the bracket if needed
- **Wrong:** Failing to apply the exponent n to the factored out constant when expanding $(a + bx)^n$
  - Why it fails: This leads to all terms in the expansion being scaled incorrectly, resulting in full loss of marks for the expansion
  - Correct: When factoring a out of $(a + bx)^n$, write it as $a^n \left(1 + \frac{b}{a}x\right)^n$, not $a \left(1 + \frac{b}{a}x\right)^n$
- **Wrong:** Using the validity condition $|x| < 1$ for positive integer n expansions
  - Why it fails: Finite expansions for positive integer n are valid for all x, so this condition is unnecessary and may cost you a mark for incorrect validity
  - Correct: Only state validity conditions for expansions where n is not a positive integer
- **Wrong:** Substituting $3x$ instead of $-3x$ into the standard formula for $(1 - 3x)^n$
  - Why it fails: This leads to incorrect signs for all odd-powered terms, losing marks even if your expansion method is otherwise correct
  - Correct: Rewrite the bracket as $(1 + (-3x))^n$ before substituting into the standard $(1 + x)^n$ formula

## Cheatsheet

| Case | Expansion Form | Validity | General r-th Term |
| --- | --- | --- | --- |
| n positive integer | $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ... + x^n$ | All real x | $\binom{n}{r}x^r$ |
| n rational (non-positive integer) | $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ...$ | $\|x\| < 1$ | $\frac{n(n-1)...(n-r+1)}{r!}x^r$ |
| $(a + bx)^n$ (non-positive integer n) | $a^n\left(1 + \frac{b}{a}x\right)^n$ | $\|x\| < \left\|\frac{a}{b}\right\|$ | $a^n \times \frac{n(n-1)...(n-r+1)}{r!}\left(\frac{b}{a}x\right)^r$ |

## What's next

Now that you have mastered the binomial series for $(1 + x)^n$, you can apply this skill to solve more complex algebra problems in your Edexcel IGCSE Further Pure Math exam. This topic is frequently combined with algebraic simplification and coordinate geometry questions, so make sure you practice connecting these skills. The next topics in the S6 unit build directly on this foundation, so you should move on to practicing past paper questions focused on binomial series applications to solidify your understanding before moving on to differentiation in S7.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s6-the-binomial-series-n/