Source: https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-4-binomial-expansion-and-rational-powers-a2-only

[Revision notes](https://senecalearning.com/en-gb/revision-notes.md) › [A Level](https://senecalearning.com/en-gb/revision-notes/a-level.md) › [Maths](https://senecalearning.com/en-gb/revision-notes/a-level/maths.md) › [Edexcel](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel.md) › Binomial Expansion & Rational Powers (A2 only)

# Binomial Expansion & Rational Powers (A2 only)

Maths: Edexcel A Level Pure Maths, topic 4.1.4

## Binomial Expansion with Rational Powers

If we have an expression $$(1+x)^n$$ and 
$$n$$ is a negative or rational number, we need to use a different equation for its expansion.

### Binomial expansion

![Binomial expansion](https://image-v2.cdn.app.senecalearning.com/2020-06/c0f797db-4a82-4865-aec4-2c2b41c35881/Screenshot%202020-06-09%20at%2014.27.34-min.png)

- The binomial expansion of a the expression $$(1+x)^n$$ is equal to:
 - $$1+ nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + ... + \frac{n(n-1)\times...\times(n-r+1)}{r!}x^r+...$$
- This is an infinite sum of terms, also known as an infinite series.
- This expansion is valid when $$|x|<1$$ and 
$$n\in$$ &#8477; (is a real number).
- This equation is included in the formula booklet in the exam.

### Binomial expansion

![Binomial expansion](https://image-v2.cdn.app.senecalearning.com/2020-06/ba9acd18-cf10-40b3-837b-9a350083ad2e/Screenshot%202020-06-09%20at%2014.27.43-min.png)

- We can use this equation to work out the expansion of binomials of the form $$(1+bx)^n$$ by grouping together 
$$(bx)$$ and treating it as if it was 
$$x$$:
 - $$(1+bx)^n=1 + n(bx) + \frac{n(n-1)}{2!}(bx)^2 + \frac{n(n-1)(n-2)}{3!}(bx)^3...$$
- This expansion is valid when $$|bx|<1$$ or 
$$|x|<\frac{\small 1}{\small |b|}$$.

### Example

![Example](https://image-v2.cdn.app.senecalearning.com/2020-06/9c8f6088-270f-45b6-adb9-5327a1e31836/Screenshot%202020-06-09%20at%2014.27.45-min.png)

- What are the first four terms of the binomial expansion of $$(1+x)^{-2}$$?
 - For which values of $$x$$ is the expansion valid?

### Comparing variables

![Comparing variables](https://image-v2.cdn.app.senecalearning.com/2020-06/4661fc03-42cc-42ff-bd2e-670874e7988d/Screenshot%202020-06-09%20at%2014.27.47-min.png)

- If we compare the expression we want to expand with the equation given in the formula booklet, we see that $$n = -2$$.
- So we can substitute this into each term to find the binomial expansion.

### First term

![First term](https://image-v2.cdn.app.senecalearning.com/2020-06/b711bf2d-0e61-425f-b94f-9e0043100a7e/Screenshot%202020-06-09%20at%2014.27.49-min.png)

- The first term of this type of binomial expansion is always equal to 1.

### Second term

![Second term](https://image-v2.cdn.app.senecalearning.com/2020-06/e36de4ee-b4ae-4d83-9ca0-ef31553d2807/Screenshot%202020-06-09%20at%2014.27.51-min.png)

- The second term of the binomial expansion is $$nx$$.
- Substituting in $$n = -2$$, we get:
 - $$nx = -2x$$

### Third term

![Third term](https://image-v2.cdn.app.senecalearning.com/2020-06/8c345911-c373-4c50-8ffd-10f81240a95c/Screenshot%202020-06-09%20at%2014.27.53-min.png)

- The third term of the binomial expansion is $$\frac{n(n-1)}{2!}x^2$$.
- Substituting in $$n = -2$$, we get:
 - $$\frac{n(n-1)}{2!}x^2 = \frac{(-2)\times(-3)}{2\times1}x^2 = 3x^2$$

### Fourth term

![Fourth term](https://image-v2.cdn.app.senecalearning.com/2020-06/875ebe83-c9f5-46f8-b17e-33961360e829/Screenshot%202020-06-09%20at%2014.27.57-min.png)

- The third term of the binomial expansion is $$\frac{n(n-1)(n-2)}{3!}x^3$$.
- Substituting in $$n = -2$$, we get:
 - $$\frac{n(n-1)(n-2)}{3!}x^3=\frac{(-2)\times(-3)\times(-4)}{3\times2\times1}x^3 = -4x^3$$

### Answer

![Answer](https://image-v2.cdn.app.senecalearning.com/2020-06/875ebe83-c9f5-46f8-b17e-33961360e829/Screenshot%202020-06-09%20at%2014.27.57-min.png)

- The first four terms of the binomial expansion of $$(1+x)^{-2}$$ are:
- $$(1+x)^{-2} = 1 -2x + 3x^2 - 4x^3 + ...$$

### Validity

![Validity](https://image-v2.cdn.app.senecalearning.com/2020-06/6f62b597-65ad-409c-a803-08ee40a0b427/Screenshot%202020-06-09%20at%2014.27.59-min.png)

- The expansion is valid for values of $$x$$ such that 
$$|x|$$<1.

## Test yourself on Binomial Expansion & Rational Powers (A2 only)

- For which values of $$x$$ is the binomial expansion of 
$$\frac{1}{1+2x}$$ valid?
- For which values of $$x$$ is the binomial expansion of 
$$(1+\frac{1}{3}x)^{-4}$$ valid?

[Practise this topic for free](https://app.senecalearning.com/classroom/course/2f7e3897-8547-4e08-b6f9-bbefc48a9d18/section/8175e80d-532c-4c2a-bd30-cdebeae37496/session)

- Previous topic: [4.1.3 Binomial Expansion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-3-binomial-expansion.md)
- Next topic: [4.1.5 Binomial Expansion & Rational Powers 2 (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-5-binomial-expansion-and-rational-powers-2-a2-only.md)

## Maths: Edexcel A Level Pure Maths contents

### 1 Proof
- 1.1 Types of Numbers
  - [1.1.1 Whole Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-1-whole-numbers.md)
  - [1.1.2 The Four Rules for Positive Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-2-the-four-rules-for-positive-numbers.md)
  - [1.1.3 Negative Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-3-negative-numbers.md)
  - [1.1.4 Integers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-4-integers.md)
  - [1.1.5 The Four Rules for Negative Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-5-the-four-rules-for-negative-numbers.md)
  - [1.1.6 Real Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-6-real-numbers.md)
  - [1.1.7 Rational vs Irrational](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-1-7-rational-vs-irrational.md)
- 1.2 Notation
  - [1.2.1 Sets](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-2-1-sets.md)
  - [1.2.2 Subsets](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-2-2-subsets.md)
  - [1.2.3 Venn Diagrams](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-2-3-venn-diagrams.md)
  - [1.2.4 Sets Numbers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-2-4-sets-numbers.md)
  - [1.2.5 Notation in Mathematical Arguments](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-2-5-notation-in-mathematical-arguments.md)
- 1.3 Proof
  - [1.3.1 Mathematical Proof](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-1-mathematical-proof.md)
  - [1.3.2 Proof by Deduction](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-2-proof-by-deduction.md)
  - [1.3.3 Proving an Identity](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-3-proving-an-identity.md)
  - [1.3.4 Polynomial Identities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-4-polynomial-identities.md)
  - [1.3.5 Proof by Exhaustion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-5-proof-by-exhaustion.md)
  - [1.3.6 Disproof by Counter Example](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-6-disproof-by-counter-example.md)
  - [1.3.7 Proof by Contradiction (A2 Only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/1-3-7-proof-by-contradiction-a2-only.md)

### 2 Algebra & Functions
- 2.1 Powers & Roots
  - [2.1.1 Exponential Rules](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-1-1-exponential-rules.md)
  - [2.1.2 Rational Exponents](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-1-2-rational-exponents.md)
  - [2.1.3 Surds](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-1-3-surds.md)
  - [2.1.4 Simplify Surds](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-1-4-simplify-surds.md)
  - [2.1.5 Rationalise Denominator](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-1-5-rationalise-denominator.md)
- 2.2 Quadratic Equations
  - [2.2.1 Quadratic +/-](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-1-quadratic.md)
  - [2.2.2 Factorise Quadratics](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-2-factorise-quadratics.md)
  - [2.2.3 Quadratic Formula](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-3-quadratic-formula.md)
  - [2.2.4 Complete the Square](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-4-complete-the-square.md)
  - [2.2.5 Solve Quadratics Graphically](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-5-solve-quadratics-graphically.md)
  - [2.2.6 Simultaneous Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-6-simultaneous-equations.md)
  - [2.2.7 Simultaneous Quadratic Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-7-simultaneous-quadratic-equations.md)
  - [2.2.8 Simultaneous Equations Graphs](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-8-simultaneous-equations-graphs.md)
  - [2.2.9 Simultaneous Equations Word Problems](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-9-simultaneous-equations-word-problems.md)
  - [2.2.10 Quadratic Equations of Powers](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-2-10-quadratic-equations-of-powers.md)
- 2.3 Inequalities
  - [2.3.1 Solving Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-1-solving-inequalities.md)
  - [2.3.2 Solving Double Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-2-solving-double-inequalities.md)
  - [2.3.3 Solving Quadratic Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-3-solving-quadratic-inequalities.md)
  - [2.3.4 Representing Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-4-representing-inequalities.md)
  - [2.3.5 Graphing Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-5-graphing-inequalities.md)
  - [2.3.6 Set Notation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-6-set-notation.md)
  - [2.3.7 Set Notation & Inequalities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-3-7-set-notation-and-inequalities.md)
- 2.4 Polynomials
  - [2.4.1 Polynomials](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-4-1-polynomials.md)
  - [2.4.2 Adding & Subtracting Polynomials](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-4-2-adding-and-subtracting-polynomials.md)
  - [2.4.3 Remainder Theorem](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-4-3-remainder-theorem.md)
  - [2.4.4 Algebraic Long Division](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-4-4-algebraic-long-division.md)
  - [2.4.5 Rewriting Rational Expressions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-4-5-rewriting-rational-expressions.md)
- 2.5 Graphs
  - [2.5.1 Graphs of Polynomials](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-1-graphs-of-polynomials.md)
  - [2.5.2 Simultaneous Equations Graphs](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-2-simultaneous-equations-graphs.md)
  - [2.5.3 Simultaneous Quadratic Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-3-simultaneous-quadratic-equations.md)
  - [2.5.4 Graphs of Proportion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-4-graphs-of-proportion.md)
  - [2.5.5 Graphs of Common Functions 1](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-5-graphs-of-common-functions-1.md)
  - [2.5.6 Graphs of Common Functions 2](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-6-graphs-of-common-functions-2.md)
  - [2.5.7 Graphs of Common Functions 3](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-7-graphs-of-common-functions-3.md)
  - [2.5.8 Proportion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-8-proportion.md)
  - [2.5.9 Direct Proportion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-9-direct-proportion.md)
  - [2.5.10 Inverse Proportion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-5-10-inverse-proportion.md)
- 2.6 Functions
  - [2.6.1 Properties of Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-6-1-properties-of-functions.md)
  - [2.6.2 Inverse Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-6-2-inverse-functions.md)
  - [2.6.3 Composite Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-6-3-composite-functions.md)
- 2.7 Transformation of Graphs
  - [2.7.1 Reflections](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-1-reflections.md)
  - [2.7.2 Rotation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-2-rotation.md)
  - [2.7.3 Translation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-3-translation.md)
  - [2.7.4 Stretches & Compressions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-4-stretches-and-compressions.md)
  - [2.7.5 Effects of Transformations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-5-effects-of-transformations.md)
  - [2.7.6 Combinations of Transformations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-7-6-combinations-of-transformations.md)
- 2.8 Partial Fractions (A2 Only)
  - [2.8.1 Partial Fractions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-8-1-partial-fractions.md)
  - [2.8.2 Partial Fractions with Three Linear Factors](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-8-2-partial-fractions-with-three-linear-factors.md)
  - [2.8.3 Partial Fractions with Three Linear Factors 2](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-8-3-partial-fractions-with-three-linear-factors-2.md)
  - [2.8.4 Partial Fractions with Repeated Factors](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-8-4-partial-fractions-with-repeated-factors.md)
  - [2.8.5 Partial Fractions - Practice](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/2-8-5-partial-fractions-practice.md)

### 3 Coordinate Geometry
- 3.1 Straight Lines
  - [3.1.1 y = mx +c](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-1-1-y-=-mx-c.md)
  - [3.1.2 Gradient & Y-intercept](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-1-2-gradient-and-y-intercept.md)
  - [3.1.3 Distance-Time Graph](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-1-3-distance-time-graph.md)
  - [3.1.4 Velocity-Time Graph](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-1-4-velocity-time-graph.md)
  - [3.1.5 Conversion Graph](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-1-5-conversion-graph.md)
- 3.2 Circles
  - [3.2.1 Equations of Circles centred at Origin](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-1-equations-of-circles-centred-at-origin.md)
  - [3.2.2 Finding the Centre & Radius](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-2-finding-the-centre-and-radius.md)
  - [3.2.3 Equation of a Tangent](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-3-equation-of-a-tangent.md)
  - [3.2.4 Circle Theorems - Perpendicular Bisector](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-4-circle-theorems-perpendicular-bisector.md)
  - [3.2.5 Circle Theorems - Angle at the Centre](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-5-circle-theorems-angle-at-the-centre.md)
  - [3.2.6 Circle Theorems - Angle at a Semi-Circle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-6-circle-theorems-angle-at-a-semi-circle.md)
  - [3.2.7 Equation of a Perpendicular Bisector](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-7-equation-of-a-perpendicular-bisector.md)
  - [3.2.8 Equation of a Circumcircle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-8-equation-of-a-circumcircle.md)
  - [3.2.9 Circumcircle of a Right-angled Triangle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-2-9-circumcircle-of-a-right-angled-triangle.md)
- 3.3 Parametric Equations (A2 only)
  - [3.3.1 Parametric Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-3-1-parametric-equations.md)
  - [3.3.2 Modelling with Parametric Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/3-3-2-modelling-with-parametric-equations.md)

### 4 Sequences & Series
- 4.1 Binomial Expansion
  - [4.1.1 Pascal's Triangle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-1-pascals-triangle.md)
  - [4.1.2 Factorials](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-2-factorials.md)
  - [4.1.3 Binomial Expansion](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-3-binomial-expansion.md)
  - [4.1.4 Binomial Expansion & Rational Powers (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-4-binomial-expansion-and-rational-powers-a2-only.md)
  - [4.1.5 Binomial Expansion & Rational Powers 2 (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-1-5-binomial-expansion-and-rational-powers-2-a2-only.md)
- 4.2 Sequences (A2 only)
  - [4.2.1 Nth term Rules](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-1-nth-term-rules.md)
  - [4.2.2 Recurring Rules](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-2-recurring-rules.md)
  - [4.2.3 Types of Sequences](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-3-types-of-sequences.md)
  - [4.2.4 Arithmetic Sequences](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-4-arithmetic-sequences.md)
  - [4.2.5 Geometric Sequences](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-5-geometric-sequences.md)
  - [4.2.6 Sigma Notation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-6-sigma-notation.md)
  - [4.2.7 Arithmetic Series](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-7-arithmetic-series.md)
  - [4.2.8 Geometric Series](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-8-geometric-series.md)
  - [4.2.9 Sum to Infinity of a Geometric Series](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-9-sum-to-infinity-of-a-geometric-series.md)
  - [4.2.10 Modelling with Sequences & Series](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/4-2-10-modelling-with-sequences-and-series.md)

### 5 Trigonometry
- 5.1 Using Angles
  - [5.1.1 Radians](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-1-1-radians.md)
  - [5.1.2 Arc length](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-1-2-arc-length.md)
  - [5.1.3 Area of Sectors](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-1-3-area-of-sectors.md)
  - [5.1.4 Area of Segments](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-1-4-area-of-segments.md)
- 5.2 Trigonometric Functions
  - [5.2.1 Unit Circle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-1-unit-circle.md)
  - [5.2.2 Properties of Trig Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-2-properties-of-trig-functions.md)
  - [5.2.3 Trigonometry](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-3-trigonometry.md)
  - [5.2.4 Exact values](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-4-exact-values.md)
  - [5.2.5 Graphs of Trigonometric Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-5-graphs-of-trigonometric-functions.md)
  - [5.2.6 Small Angle Approximation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-2-6-small-angle-approximation.md)
- 5.3 Triangle Rules
  - [5.3.1 Sine Rules](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-3-1-sine-rules.md)
  - [5.3.2 Cosine Rules](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-3-2-cosine-rules.md)
  - [5.3.3 Area of a Triangle](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-3-3-area-of-a-triangle.md)
- 5.4 Trigonometric Identities
  - [5.4.1 Secant, Cosecant & Cotangent](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-4-1-secant-cosecant-and-cotangent.md)
  - [5.4.2 Pythagorean Identities](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-4-2-pythagorean-identities.md)
  - [5.4.3 Angle Addition Formulae](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-4-3-angle-addition-formulae.md)
  - [5.4.4 Manipulating the Angle Addition Formulae](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-4-4-manipulating-the-angle-addition-formulae.md)
  - [5.4.5 Double Angle Formula](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/5-4-5-double-angle-formula.md)

### 6 Exponentials & Logarithms
- 6.1 Exponentials & Logarithms
  - [6.1.1 Exponential Function](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-1-1-exponential-function.md)
  - [6.1.2 Logarithms](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-1-2-logarithms.md)
  - [6.1.3 Properties of Logarithms](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-1-3-properties-of-logarithms.md)
  - [6.1.4 Natural Logarithm](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-1-4-natural-logarithm.md)
  - [6.1.5 Graphs of Exponents & Logarithms](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-1-5-graphs-of-exponents-and-logarithms.md)
- 6.2 Exponential Growth  Decay
  - [6.2.1 Population Growth](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-1-population-growth.md)
  - [6.2.2 Depreciation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-2-depreciation.md)
  - [6.2.3 Graphing Linear and Exponential Growth](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-3-graphing-linear-and-exponential-growth.md)
  - [6.2.4 Interpreting Exponential Growth and Decay Graphs](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-4-interpreting-exponential-growth-and-decay-graphs.md)
  - [6.2.5 Compound Growth & Decay](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-5-compound-growth-and-decay.md)
  - [6.2.6 Compund Interest](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/6-2-6-compund-interest.md)

### 7 Differentiation
- 7.1 Derivatives
  - [7.1.1 Gradient of a Curve](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-1-gradient-of-a-curve.md)
  - [7.1.2 Differentiation from First Principles](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-2-differentiation-from-first-principles.md)
  - [7.1.3 Differentiation from First Principles - Practice](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-3-differentiation-from-first-principles-practice.md)
  - [7.1.4 Differentiation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-4-differentiation.md)
  - [7.1.5 Finding derivatives](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-5-finding-derivatives.md)
  - [7.1.6 Second Order Derivatives](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-1-6-second-order-derivatives.md)
- 7.2 Graphs & Differentiation
  - [7.2.1 Stationary Points](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-2-1-stationary-points.md)
  - [7.2.2 Maxima & Minima](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-2-2-maxima-and-minima.md)
  - [7.2.3 Modelling with Differentiation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-2-3-modelling-with-differentiation.md)
  - [7.2.4 Using Second Derivatives (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-2-4-using-second-derivatives-a2-only.md)
  - [7.2.5 Rate of Change (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-2-5-rate-of-change-a2-only.md)
- 7.3 Differentiation With Trigonometry and Exponentials
  - [7.3.1 Differentiation with Sin & Cos](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-3-1-differentiation-with-sin-and-cos.md)
  - [7.3.2 Diff. of Exponential Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-3-2-diff-of-exponential-functions.md)
  - [7.3.3 Differentiation of Trig. Functions 1 (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-3-3-differentiation-of-trig-functions-1-a2-only.md)
  - [7.3.4 Differentiation of Trig. Functions 2 (A2 only)](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-3-4-differentiation-of-trig-functions-2-a2-only.md)
- 7.4 Rules of Differetiation (A2 only)
  - [7.4.1 Chain Rule](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-4-1-chain-rule.md)
  - [7.4.2 Product Rule](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-4-2-product-rule.md)
  - [7.4.3 Quotient Rule](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-4-3-quotient-rule.md)
- 7.5 Parametric & Implicit Differentiation
  - [7.5.1 Parametric Differentiation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-5-1-parametric-differentiation.md)
  - [7.5.2 Implicit Differentiation](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/7-5-2-implicit-differentiation.md)

### 8 Integration
- 8.1 Integration
  - [8.1.1 Integration - Basics](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-1-integration-basics.md)
  - [8.1.2 Integration - Definite Integrals](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-2-integration-definite-integrals.md)
  - [8.1.3 Finding Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-3-finding-functions.md)
  - [8.1.4 Area under curves](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-4-area-under-curves.md)
  - [8.1.5 Area under x-axis](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-5-area-under-x-axis.md)
  - [8.1.6 Area between curves and lines](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-1-6-area-between-curves-and-lines.md)
- 8.2 Integration (A2 Only)
  - [8.2.1 Integration - Standard Functions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-1-integration-standard-functions.md)
  - [8.2.2 Integration - Reverse Chain Rule](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-2-integration-reverse-chain-rule.md)
  - [8.2.3 Integration by Substitution](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-3-integration-by-substitution.md)
  - [8.2.4 Integration by Parts](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-4-integration-by-parts.md)
  - [8.2.5 Integration by Parts - Practice](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-5-integration-by-parts-practice.md)
  - [8.2.6 Integration by Partial Fraction](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-6-integration-by-partial-fraction.md)
  - [8.2.7 Modelling with Differential Equations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/8-2-7-modelling-with-differential-equations.md)

### 9 Numerical Methods
- 9.1 Finding Solutions
  - [9.1.1 Number Iterations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-1-1-number-iterations.md)
  - [9.1.2 Iterations to 1 Decimal Places](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-1-2-iterations-to-1-decimal-places.md)
  - [9.1.3 Iteration Formula](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-1-3-iteration-formula.md)
  - [9.1.4 Locating Roots](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-1-4-locating-roots.md)
  - [9.1.5 Newton Raphson Method](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-1-5-newton-raphson-method.md)
- 9.2 Finding the Area
  - [9.2.1 Trapezium Rule](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/9-2-1-trapezium-rule.md)

### 10 Vectors
- 10.1 2D Vectors
  - [10.1.1 Vectors - Magnitude & Direction](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-1-vectors-magnitude-and-direction.md)
  - [10.1.2 Unit Vectors](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-2-unit-vectors.md)
  - [10.1.3 Vectors - Algebraic Operations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-3-vectors-algebraic-operations.md)
  - [10.1.4 Position Vectors](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-4-position-vectors.md)
  - [10.1.5 Vector Proofs](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-5-vector-proofs.md)
  - [10.1.6 Vectors Word Questions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-1-6-vectors-word-questions.md)
- 10.2 3D Vectors
  - [10.2.1 3D Vector Magnitude](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-2-1-3d-vector-magnitude.md)
  - [10.2.2 3D Vectors - Algebraic Operations](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-2-2-3d-vectors-algebraic-operations.md)
  - [10.2.3 Position Vectors 3D](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-2-3-position-vectors-3d.md)
  - [10.2.4 3D Vectors Word Questions](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-2-4-3d-vectors-word-questions.md)
- 10.3 Vector Proofs
  - [10.3.1 Coming Soon](https://senecalearning.com/en-gb/revision-notes/a-level/maths/edexcel/10-3-1-coming-soon.md)