# Binomial Expansion

> Edexcel International A-Level Mathematics · IAL 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p4-binomial-expansion/

This guide covers the Edexcel IAL P4 binomial series for any rational exponent, including rescaling to expand $(ax+b)^n$, stating validity intervals, and expanding rational functions using partial fraction decomposition.

**Prerequisites:** [Partial fraction decomposition (P4 §2.1)](https://www.owlsprep.com/study/edexcel-ial-math-p4-partial-fractions/); [Basic algebraic manipulation](https://www.owlsprep.com/study/edexcel-ial-math-p2-algebraic-manipulation/)

## Learning objectives

- Derive the binomial expansion of (ax+b)^n for any rational n
- State the correct validity interval for any binomial series
- Expand rational functions by first decomposing into partial fractions
- Apply expansions to compute valid approximations for small x values

## Rescaling the Standard Binomial Series

The formula booklet provides the standard binomial series for $(1+x)^n$ where $n$ is any rational number:

$$1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$$

To expand $(ax+b)^n$, first factor out the constant term $b$ to rewrite the expression in the form $1 + kx$:

$$(ax + b)^n = b^n\left(1 + \frac{a}{b}x\right)^n$$

**Binomial Series for Rational n** — An infinite power series expansion of a binomial raised to a rational exponent, convergent only within a specific interval of x values

*Example:* $(1 - 3x)^{-1} = 1 + 3x + 9x^2 + 27x^3 + \dots$ for $|x|<\frac{1}{3}$

**Worked example:** Find the first three non-zero terms of the expansion of $(2 + 4x)^{-\frac{1}{2}}$

1. 1. Factor out the constant term 2 from the bracket:

   $$(2 + 4x)^{-\frac{1}{2}} = 2^{-\frac{1}{2}}\left(1 + \frac{4}{2}x\right)^{-\frac{1}{2}} = \frac{1}{\sqrt{2}}\left(1 + 2x\right)^{-\frac{1}{2}}$$
2. 2. Apply the standard binomial series to $(1 + 2x)^{-\frac{1}{2}}$ with $n = -\frac{1}{2}$, replacing $x$ with $2x$:

   $$(1 + 2x)^{-\frac{1}{2}} = 1 + \left(-\frac{1}{2}\right)(2x) + \frac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2!}(2x)^2 + \dots$$
3. 3. Simplify each term:

   $$= 1 - x + \frac{3}{8}(4x^2) + \dots = 1 - x + \frac{3}{2}x^2 + \dots$$
4. 4. Multiply through by the factored constant $\frac{1}{\sqrt{2}}$:

   $$= \frac{1}{\sqrt{2}} - \frac{x}{\sqrt{2}} + \frac{3x^2}{2\sqrt{2}} + \dots (rationalise if required)$$

> **Exam tip:** Always factor out the larger constant term first to ensure the remaining bracket is in the form $1 + kx$, avoiding arithmetic errors.

*Calculator:* allowed

## Validity Intervals for Binomial Series

The standard binomial series only converges when the argument of the $1 + kx$ bracket is less than 1 in absolute value. For the expanded form $b^n\left(1 + \frac{a}{b}x\right)^n$, the convergence condition is:

$$\left|\frac{a}{b}x\right| < 1$$

Rearranging gives the validity interval for x: $|x| < \frac{b}{a}$. You must always state this interval alongside any binomial expansion to gain full marks.

> **warning**
>
> If you expand multiple terms from partial fractions, the overall validity interval is the intersection (smallest interval) of the individual validity intervals for each term.

**Worked example:** State the validity interval for the expansion of $(3 - 6x)^{\frac{2}{3}}$

1. 1. Factor out the constant term 3:

   $$(3 - 6x)^{\frac{2}{3}} = 3^{\frac{2}{3}}\left(1 - 2x\right)^{\frac{2}{3}}$$
2. 2. Apply the convergence condition to the bracket:

   $$|-2x| < 1 \implies 2|x| < 1$$
3. 3. Rearrange to solve for $|x|$:

   $$|x| < \frac{1}{2}$$

**Check your understanding**

Test your understanding of validity intervals:

1. What is the validity interval for $(4 + 12x)^{-2}$?

   - $|x| < 1/3$
   - $|x| < 3$
   - $|x| < 1/4$
   - $|x| < 4$

   *Answer:* $|x| < 1/3$

   *Why:* Correct: factoring out 4 gives $(1 + 3x)^{-2}$, so $|3x| < 1 \implies |x| < 1/3$.

*Calculator:* allowed

## Expanding Rational Functions via Partial Fractions

Many exam questions ask you to expand rational functions where the numerator has degree less than the denominator. First decompose the function into partial fractions (from P4 §2.1), then expand each partial fraction term separately using the binomial series, then combine like terms.

**Worked example:** Given $\frac{3x + 5}{(x+1)(x+2)}$, find the first three terms of its binomial expansion and state the overall validity interval.

1. 1. First decompose into partial fractions and rewrite in binomial form:

   $$\frac{3x +5}{(x+1)(x+2)} = \frac{2}{x+1} + \frac{1}{x+2} = 2(1 + x)^{-1} + \frac{1}{2}\left(1 + \frac{x}{2}\right)^{-1}$$
2. 2. Expand each term separately: 
- First term: $2(1+x)^{-1} = 2\left(1 - x + x^2 + \dots\right) = 2 - 2x + 2x^2 + \dots$, valid for $|x|<1$ 
- Second term: $\frac{1}{2}\left(1 + \frac{x}{2}\right)^{-1} = \frac{1}{2}\left(1 - \frac{x}{2} + \frac{x^2}{4} + \dots\right) = \frac{1}{2} - \frac{x}{4} + \frac{x^2}{8} + \dots$, valid for $|x|<2$
3. 3. Combine like terms:

   $$(2 + 1/2) + (-2x - x/4) + (2x^2 + x^2/8) + \dots = \frac{5}{2} - \frac{9x}{4} + \frac{17x^2}{8} + \dots$$
4. 4. Overall validity interval is the smaller of $|x|<1$ and $|x|<2$, so $|x|<1$.

> **Exam tip:** Always decompose first before expanding; do not attempt to expand the entire denominator directly as this will lead to errors and lost marks.

*Calculator:* allowed

## Approximations Using Binomial Expansions

You may be asked to use your binomial expansion to approximate values of the original expression for a small x within the validity interval. Substitute the small x value into the first few terms of the expansion to get an approximate value.

**Worked example:** Use the first three terms of the expansion of $(1 + 0.02)^{-3}$ to find an approximate value, giving your answer to 4 decimal places.

1. 1. Write the expansion of $(1 + x)^{-3}$ up to the $x^2$ term:

   $$(1+x)^{-3} = 1 - 3x + \frac{(-3)(-4)}{2!}x^2 + \dots = 1 - 3x + 6x^2 + \dots$$
2. 2. Substitute $x=0.02$ into the expansion:

   $$(1 + 0.02)^{-3} ≈ 1 - 3(0.02) + 6(0.02)^2 = 1 - 0.06 + 0.0024 = 0.9424$$
3. 3. Verify the value is within the validity interval: $|0.02| < 1$, so the approximation is valid.

> **Exam tip:** Always check the x value is within the validity interval before using the expansion for approximation; if it is not, the approximation will be incorrect.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using the positive integer $n$ binomial formula with $^nC_r$ for rational exponents
  - Why it fails: The $^nC_r$ formula only works for finite expansions when $n$ is a positive integer, not for infinite series with rational exponents
  - Correct: Use the standard infinite series formula given in the formula booklet for all rational exponents in P4
- **Wrong:** Forgetting to factor out the constant term before applying the standard series
  - Why it fails: The standard series only applies to expressions of the form $(1 + kx)$, so applying it directly to $(ax + b)$ gives incorrect terms
  - Correct: Always factor out the constant $b$ first to rewrite as $b^n\left(1 + \frac{a}{b}x\right)^n$ before expanding
- **Wrong:** Stating the validity interval as $|x| < a/b$ instead of $|x|<b/a$
  - Why it fails: The convergence condition requires $|(a/b)x| < 1$, rearranged gives $|x| < b/a$, reversing the fraction leads to an invalid interval
  - Correct: Rearrange the convergence condition carefully to solve for $|x|$, checking your algebra each time
- **Wrong:** Taking the union of validity intervals for partial fraction expansions instead of the intersection
  - Why it fails: The series only converges when all individual terms converge, so you need the interval that satisfies all validity conditions
  - Correct: Select the smallest of the individual validity intervals as the overall interval for the combined expansion

## Cheatsheet

| Expression | Expansion Method | Validity Interval |
| --- | --- | --- |
| $(1+x)^n$, $n$ rational | Use given formula: $1 + nx + \frac{n(n-1)}{2!}x^2 + \dots$ | $\|x\| < 1$ |
| $(ax + b)^n$, $n$ rational | Factor out $b^n$, expand $\left(1 + \frac{a}{b}x\right)^n$ via standard formula | $\|x\| < \frac{b}{a}$ |
| Rational function | Decompose into partial fractions first, expand each term separately, combine like terms | Smallest interval of individual partial fraction validity intervals |

## What's next

Now that you have mastered binomial expansions for Edexcel IAL P4, you can apply this skill to other topics in the unit, including differentiation of power series and numerical approximations. This topic is frequently tested alongside partial fractions and algebraic manipulation, so make sure you practice past paper questions to solidify your understanding before your exam. The skills you have learned here will also be useful if you go on to study Further Mathematics, where binomial series are extended to complex exponents.

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