4.1.3

Binomial Expansion

Test yourself on Binomial Expansion

Test your knowledge with free interactive questions on Seneca — used by over 10 million students.

Binomial Expansion

We can work out the expansion of binomials using an equation involving factorials.

$Expanding binomials$

Expanding binomials

We have seen how we can expand binomials using the numbers in each row of Pascal's triangle.
We've also seen how to work out the numbers in Pascal's triangle using factorials.
This leads us to an equation we can use to expand binomials without having to think about Pascal's triangle.
- This is called the binomial expansion or binomial theorem.

Binomial expansion equation

Binomial expansion equation

The binomial expansion of $(a+b)^n$ is:
- $(a+b)^n=a^n + {n\choose 1}a^{n-1}b + {n\choose 2}a^{n-2}b^2 + ...+{n \choose r}a^{n-r}b^{r}+...+ b^n$
This equation applies when $n \in$ ℕ (is a positive whole number).
This equation is included in the formula booklet in the exam.

General term

General term

The general term in a binomial expansion is given by ${n \choose r}a^{n-r}b^{r}$ .
We can use this to find coefficients of specific orders of variables in the binomial expansion.

Example

Example

Use the binomial theorem to find the expansion of $(1-6x)^5$ .

Comparing variables

Comparing variables

If we compare the expression we want to expand with the equation given in the formula booklet:
- $a = 1, b = -6x$ and $n = 5$ .
So we can substitute these into each term to find the binomial expansion.

First term

First term

The first term of the binomial expansion is $a^n$ .
Substituting in $a = 1$ and $n = 5$ , we have:
- $a^n = 1^5 = 1$

Second term

Second term

The second term of the binomial expansion is ${n\choose 1}a^{n-1}b$ .
Substituting in for $a,b$ and $n$ , we have:
- ${n\choose 1}a^{n-1}b = {5\choose 1}1^4(-6x)$
Simplifying this term gives:
- $\frac{5!}{1!\times4!}\times(-6x) = -30x$

Third term

Third term

The third term of the binomial expansion is ${n\choose 2}a^{n-2}b^2$ .
Substituting in for $a,b$ and $n$ , we have:
- ${n\choose 2}a^{n-2}b^2={5\choose 2}1^{3}(-6x)^2$
Simplifying this term gives:
- $\frac{5!}{2!\times3!}\times36x^2 = 10\times36x^2 = 360x^2$

Fourth term

Fourth term

The fourth term of the binomial expansion is ${n\choose 3}a^{n-3}b^3$ .
Substituting in for $a,b$ and $n$ , we have:
- ${n\choose 3}a^{n-3}b^3={5\choose 3}1^{2}(-6x)^3$
Simplifying this term gives:
- $\frac{5!}{3!\times2!}\times(-216)x^3 = 10\times(-216)x^3 = -2160x^3$

Fifth term

Fifth term

The fifth term of the binomial expansion is ${n\choose 4}a^{n-4}b^4$ .
Substituting in for $a,b$ and $n$ , we have:
- ${n\choose 4}a^{n-4}b^4 = {5\choose 4}ab^4$
Simplifying this term gives:
- $\frac{5!}{4!\times1!}\times1296x^4 = 5\times1296x^4 = 6480x^4$

Final term

Final term

The sixth term of the binomial expansion is ${n\choose 5}a^{n-5}b^5$ .
- We can tell that this is the final term in the expansion because there will be no powers of $a$ left in the term and that ${5\choose 5}=1$ .
So this term is simply equal to $b^n$ .
Substituting in for $b$ and $n$ , we have:
- $b^n = (-6x)^5 = -7776x^5$

Sum the terms

Sum the terms

The expansion of $(1-6x)^5$ is equal to the sum of the terms in the binomial expansion, so the answer is:
- $(1-6x)^5 = 1 -30x + 360x^2 -2160x^3 + 6480x^4-7776x^5$

1

Proof

1.1

Types of Numbers

1.1.1Whole Numbers 1.1.2The Four Rules for Positive Numbers 1.1.3Negative Numbers 1.1.4Integers 1.1.5The Four Rules for Negative Numbers 1.1.6Real Numbers 1.1.7Rational vs Irrational

1.2

Notation

1.2.1Sets 1.2.2Subsets 1.2.3Venn Diagrams 1.2.4Sets Numbers 1.2.5Notation in Mathematical Arguments

1.3

Proof

1.3.1Mathematical Proof 1.3.2Proof by Deduction 1.3.3Proving an Identity 1.3.4Polynomial Identities 1.3.5Proof by Exhaustion 1.3.6Disproof by Counter Example 1.3.7Proof by Contradiction (A2 Only)

2

Algebra & Functions

2.1

Powers & Roots

2.1.1Exponential Rules 2.1.2Rational Exponents 2.1.3Surds 2.1.4Simplify Surds 2.1.5Rationalise Denominator

2.2

Quadratic Equations

2.2.1Quadratic +/-2.2.2Factorise Quadratics 2.2.3Quadratic Formula 2.2.4Complete the Square 2.2.5Solve Quadratics Graphically 2.2.6Simultaneous Equations 2.2.7Simultaneous Quadratic Equations 2.2.8Simultaneous Equations Graphs 2.2.9Simultaneous Equations Word Problems 2.2.10Quadratic Equations of Powers

2.3

Inequalities

2.3.1Solving Inequalities 2.3.2Solving Double Inequalities 2.3.3Solving Quadratic Inequalities 2.3.4Representing Inequalities 2.3.5Graphing Inequalities 2.3.6Set Notation 2.3.7Set Notation & Inequalities

2.4

Polynomials

2.4.1Polynomials 2.4.2Adding & Subtracting Polynomials 2.4.3Remainder Theorem 2.4.4Algebraic Long Division 2.4.5Rewriting Rational Expressions

2.5

Graphs

2.5.1Graphs of Polynomials 2.5.2Simultaneous Equations Graphs 2.5.3Simultaneous Quadratic Equations 2.5.4Graphs of Proportion 2.5.5Graphs of Common Functions 1 2.5.6Graphs of Common Functions 2 2.5.7Graphs of Common Functions 3 2.5.8Proportion 2.5.9Direct Proportion 2.5.10Inverse Proportion

2.6

Functions

2.6.1Properties of Functions 2.6.2Inverse Functions 2.6.3Composite Functions

2.7

Transformation of Graphs

2.7.1Reflections 2.7.2Rotation 2.7.3Translation 2.7.4Stretches & Compressions 2.7.5Effects of Transformations 2.7.6Combinations of Transformations

2.8

Partial Fractions (A2 Only)

2.8.1Partial Fractions 2.8.2Partial Fractions with Three Linear Factors 2.8.3Partial Fractions with Three Linear Factors 2 2.8.4Partial Fractions with Repeated Factors 2.8.5Partial Fractions - Practice

3

Coordinate Geometry

3.1

Straight Lines

3.1.1y = mx +c 3.1.2Gradient & Y-intercept 3.1.3Distance-Time Graph 3.1.4Velocity-Time Graph 3.1.5Conversion Graph

3.2

Circles

3.2.1Equations of Circles centred at Origin 3.2.2Finding the Centre & Radius 3.2.3Equation of a Tangent 3.2.4Circle Theorems - Perpendicular Bisector 3.2.5Circle Theorems - Angle at the Centre 3.2.6Circle Theorems - Angle at a Semi-Circle 3.2.7Equation of a Perpendicular Bisector 3.2.8Equation of a Circumcircle 3.2.9Circumcircle of a Right-angled Triangle

3.3

Parametric Equations (A2 only)

3.3.1Parametric Equations 3.3.2Modelling with Parametric Equations

4

Sequences & Series

4.1

Binomial Expansion

4.1.1Pascal's Triangle 4.1.2Factorials 4.1.3Binomial Expansion 4.1.4Binomial Expansion & Rational Powers (A2 only)4.1.5Binomial Expansion & Rational Powers 2 (A2 only)

4.2

Sequences (A2 only)

4.2.1Nth term Rules 4.2.2Recurring Rules 4.2.3Types of Sequences 4.2.4Arithmetic Sequences 4.2.5Geometric Sequences 4.2.6Sigma Notation 4.2.7Arithmetic Series 4.2.8Geometric Series 4.2.9Sum to Infinity of a Geometric Series 4.2.10Modelling with Sequences & Series

5

Trigonometry

5.1

Using Angles

5.1.1Radians 5.1.2Arc length 5.1.3Area of Sectors 5.1.4Area of Segments

5.2

Trigonometric Functions

5.2.1Unit Circle 5.2.2Properties of Trig Functions 5.2.3Trigonometry 5.2.4Exact values 5.2.5Graphs of Trigonometric Functions 5.2.6Small Angle Approximation

5.3

Triangle Rules

5.3.1Sine Rules 5.3.2Cosine Rules 5.3.3Area of a Triangle

5.4

Trigonometric Identities

5.4.1Secant, Cosecant & Cotangent 5.4.2Pythagorean Identities 5.4.3Angle Addition Formulae 5.4.4Manipulating the Angle Addition Formulae 5.4.5Double Angle Formula

6

Exponentials & Logarithms

6.1

Exponentials & Logarithms

6.1.1Exponential Function 6.1.2Logarithms 6.1.3Properties of Logarithms 6.1.4Natural Logarithm 6.1.5Graphs of Exponents & Logarithms

6.2

Exponential Growth Decay

6.2.1Population Growth 6.2.2Depreciation 6.2.3Graphing Linear and Exponential Growth 6.2.4Interpreting Exponential Growth and Decay Graphs 6.2.5Compound Growth & Decay 6.2.6Compund Interest

7

Differentiation

7.1

Derivatives

7.1.1Gradient of a Curve 7.1.2Differentiation from First Principles 7.1.3Differentiation from First Principles - Practice 7.1.4Differentiation 7.1.5Finding derivatives 7.1.6Second Order Derivatives

7.2

Graphs & Differentiation

7.2.1Stationary Points 7.2.2Maxima & Minima 7.2.3Modelling with Differentiation 7.2.4Using Second Derivatives (A2 only)7.2.5Rate of Change (A2 only)

7.3

Differentiation With Trigonometry and Exponentials

7.3.1Differentiation with Sin & Cos 7.3.2Diff. of Exponential Functions 7.3.3Differentiation of Trig. Functions 1 (A2 only)7.3.4Differentiation of Trig. Functions 2 (A2 only)

7.4

Rules of Differetiation (A2 only)

7.4.1Chain Rule 7.4.2Product Rule 7.4.3Quotient Rule

7.5

Parametric & Implicit Differentiation

7.5.1Parametric Differentiation 7.5.2Implicit Differentiation

8

Integration

8.1

Integration

8.1.1Integration - Basics 8.1.2Integration - Definite Integrals 8.1.3Finding Functions 8.1.4Area under curves 8.1.5Area under x-axis 8.1.6Area between curves and lines

8.2

Integration (A2 Only)

8.2.1Integration - Standard Functions 8.2.2Integration - Reverse Chain Rule 8.2.3Integration by Substitution 8.2.4Integration by Parts 8.2.5Integration by Parts - Practice 8.2.6Integration by Partial Fraction 8.2.7Modelling with Differential Equations

9

Numerical Methods

9.1

Finding Solutions

9.1.1Number Iterations 9.1.2Iterations to 1 Decimal Places 9.1.3Iteration Formula 9.1.4Locating Roots 9.1.5Newton Raphson Method

9.2

Finding the Area

9.2.1Trapezium Rule

10

Vectors

10.1

2D Vectors

10.1.1Vectors - Magnitude & Direction 10.1.2Unit Vectors 10.1.3Vectors - Algebraic Operations 10.1.4Position Vectors 10.1.5Vector Proofs 10.1.6Vectors Word Questions

10.2

3D Vectors

10.2.13D Vector Magnitude 10.2.23D Vectors - Algebraic Operations 10.2.3Position Vectors 3D 10.2.43D Vectors Word Questions

10.3

Vector Proofs

10.3.1Coming Soon

Jump to other topics

1

Proof

1.1

Types of Numbers

1.1.1Whole Numbers 1.1.2The Four Rules for Positive Numbers 1.1.3Negative Numbers 1.1.4Integers 1.1.5The Four Rules for Negative Numbers 1.1.6Real Numbers 1.1.7Rational vs Irrational

1.2

Notation

1.2.1Sets 1.2.2Subsets 1.2.3Venn Diagrams 1.2.4Sets Numbers 1.2.5Notation in Mathematical Arguments

1.3

Proof

1.3.1Mathematical Proof 1.3.2Proof by Deduction 1.3.3Proving an Identity 1.3.4Polynomial Identities 1.3.5Proof by Exhaustion 1.3.6Disproof by Counter Example 1.3.7Proof by Contradiction (A2 Only)

2

Algebra & Functions

2.1

Powers & Roots

2.1.1Exponential Rules 2.1.2Rational Exponents 2.1.3Surds 2.1.4Simplify Surds 2.1.5Rationalise Denominator

2.2

Quadratic Equations

2.2.1Quadratic +/-2.2.2Factorise Quadratics 2.2.3Quadratic Formula 2.2.4Complete the Square 2.2.5Solve Quadratics Graphically 2.2.6Simultaneous Equations 2.2.7Simultaneous Quadratic Equations 2.2.8Simultaneous Equations Graphs 2.2.9Simultaneous Equations Word Problems 2.2.10Quadratic Equations of Powers

2.3

Inequalities

2.3.1Solving Inequalities 2.3.2Solving Double Inequalities 2.3.3Solving Quadratic Inequalities 2.3.4Representing Inequalities 2.3.5Graphing Inequalities 2.3.6Set Notation 2.3.7Set Notation & Inequalities

2.4

Polynomials

2.4.1Polynomials 2.4.2Adding & Subtracting Polynomials 2.4.3Remainder Theorem 2.4.4Algebraic Long Division 2.4.5Rewriting Rational Expressions

2.5

Graphs

2.5.1Graphs of Polynomials 2.5.2Simultaneous Equations Graphs 2.5.3Simultaneous Quadratic Equations 2.5.4Graphs of Proportion 2.5.5Graphs of Common Functions 1 2.5.6Graphs of Common Functions 2 2.5.7Graphs of Common Functions 3 2.5.8Proportion 2.5.9Direct Proportion 2.5.10Inverse Proportion

2.6

Functions

2.6.1Properties of Functions 2.6.2Inverse Functions 2.6.3Composite Functions

2.7

Transformation of Graphs

2.7.1Reflections 2.7.2Rotation 2.7.3Translation 2.7.4Stretches & Compressions 2.7.5Effects of Transformations 2.7.6Combinations of Transformations

2.8

Partial Fractions (A2 Only)

2.8.1Partial Fractions 2.8.2Partial Fractions with Three Linear Factors 2.8.3Partial Fractions with Three Linear Factors 2 2.8.4Partial Fractions with Repeated Factors 2.8.5Partial Fractions - Practice

3

Coordinate Geometry

3.1

Straight Lines

3.1.1y = mx +c 3.1.2Gradient & Y-intercept 3.1.3Distance-Time Graph 3.1.4Velocity-Time Graph 3.1.5Conversion Graph

3.2

Circles

3.2.1Equations of Circles centred at Origin 3.2.2Finding the Centre & Radius 3.2.3Equation of a Tangent 3.2.4Circle Theorems - Perpendicular Bisector 3.2.5Circle Theorems - Angle at the Centre 3.2.6Circle Theorems - Angle at a Semi-Circle 3.2.7Equation of a Perpendicular Bisector 3.2.8Equation of a Circumcircle 3.2.9Circumcircle of a Right-angled Triangle

3.3

Parametric Equations (A2 only)

3.3.1Parametric Equations 3.3.2Modelling with Parametric Equations

4

Sequences & Series

4.1

Binomial Expansion

4.1.1Pascal's Triangle 4.1.2Factorials 4.1.3Binomial Expansion 4.1.4Binomial Expansion & Rational Powers (A2 only)4.1.5Binomial Expansion & Rational Powers 2 (A2 only)

4.2

Sequences (A2 only)

4.2.1Nth term Rules 4.2.2Recurring Rules 4.2.3Types of Sequences 4.2.4Arithmetic Sequences 4.2.5Geometric Sequences 4.2.6Sigma Notation 4.2.7Arithmetic Series 4.2.8Geometric Series 4.2.9Sum to Infinity of a Geometric Series 4.2.10Modelling with Sequences & Series

5

Trigonometry

5.1

Using Angles

5.1.1Radians 5.1.2Arc length 5.1.3Area of Sectors 5.1.4Area of Segments

5.2

Trigonometric Functions

5.2.1Unit Circle 5.2.2Properties of Trig Functions 5.2.3Trigonometry 5.2.4Exact values 5.2.5Graphs of Trigonometric Functions 5.2.6Small Angle Approximation

5.3

Triangle Rules

5.3.1Sine Rules 5.3.2Cosine Rules 5.3.3Area of a Triangle

5.4

Trigonometric Identities

5.4.1Secant, Cosecant & Cotangent 5.4.2Pythagorean Identities 5.4.3Angle Addition Formulae 5.4.4Manipulating the Angle Addition Formulae 5.4.5Double Angle Formula

6

Exponentials & Logarithms

6.1

Exponentials & Logarithms

6.1.1Exponential Function 6.1.2Logarithms 6.1.3Properties of Logarithms 6.1.4Natural Logarithm 6.1.5Graphs of Exponents & Logarithms

6.2

Exponential Growth Decay

6.2.1Population Growth 6.2.2Depreciation 6.2.3Graphing Linear and Exponential Growth 6.2.4Interpreting Exponential Growth and Decay Graphs 6.2.5Compound Growth & Decay 6.2.6Compund Interest

7

Differentiation

7.1

Derivatives

7.1.1Gradient of a Curve 7.1.2Differentiation from First Principles 7.1.3Differentiation from First Principles - Practice 7.1.4Differentiation 7.1.5Finding derivatives 7.1.6Second Order Derivatives

7.2

Graphs & Differentiation

7.2.1Stationary Points 7.2.2Maxima & Minima 7.2.3Modelling with Differentiation 7.2.4Using Second Derivatives (A2 only)7.2.5Rate of Change (A2 only)

7.3

Differentiation With Trigonometry and Exponentials

7.3.1Differentiation with Sin & Cos 7.3.2Diff. of Exponential Functions 7.3.3Differentiation of Trig. Functions 1 (A2 only)7.3.4Differentiation of Trig. Functions 2 (A2 only)

7.4

Rules of Differetiation (A2 only)

7.4.1Chain Rule 7.4.2Product Rule 7.4.3Quotient Rule

7.5

Parametric & Implicit Differentiation

7.5.1Parametric Differentiation 7.5.2Implicit Differentiation

8

Integration

8.1

Integration

8.1.1Integration - Basics 8.1.2Integration - Definite Integrals 8.1.3Finding Functions 8.1.4Area under curves 8.1.5Area under x-axis 8.1.6Area between curves and lines

8.2

Integration (A2 Only)

8.2.1Integration - Standard Functions 8.2.2Integration - Reverse Chain Rule 8.2.3Integration by Substitution 8.2.4Integration by Parts 8.2.5Integration by Parts - Practice 8.2.6Integration by Partial Fraction 8.2.7Modelling with Differential Equations

9

Numerical Methods

9.1

Finding Solutions

9.1.1Number Iterations 9.1.2Iterations to 1 Decimal Places 9.1.3Iteration Formula 9.1.4Locating Roots 9.1.5Newton Raphson Method

9.2

Finding the Area

9.2.1Trapezium Rule

10

Vectors

10.1

2D Vectors

10.1.1Vectors - Magnitude & Direction 10.1.2Unit Vectors 10.1.3Vectors - Algebraic Operations 10.1.4Position Vectors 10.1.5Vector Proofs 10.1.6Vectors Word Questions

10.2

3D Vectors

10.2.13D Vector Magnitude 10.2.23D Vectors - Algebraic Operations 10.2.3Position Vectors 3D 10.2.43D Vectors Word Questions

10.3

Vector Proofs

10.3.1Coming Soon

Practice questions on Binomial Expansion

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

1
For which values of $$n$$ does the following binomial expansion apply? <br>Multiple choice

Answer all questions on Binomial Expansion

Unlock your full potential with Seneca Premium

Unlimited access to 10,000+ open-ended exam questions
Mini-mock exams based on your study history
Unlock 800+ premium courses & e-books

Get started with Seneca Premium

Factorials

Binomial Expansion & Rational Powers (A2 only)