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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 1. Number & Algebra / 1.6 Binomial Theorem / 1.6.1 Binomial Theorem


1.6.1 Binomial Theorem


Binomial Theorem

What is the Binomial Theorem?

  • The binomial theorem (sometimes known as the binomial expansion) gives a method for expanding a two-term expression in a bracket raised to a power
    • A binomial expression is in fact any two terms inside the bracket, however in IB the expression will usually be linear
  • To expand a bracket with a two-term expression in:
    • First choose the most appropriate parts of the expression to assign to a and b
    • Then use the formula for the binomial theorem:

open parentheses a plus b close parentheses to the power of n equals a to the power of n plus blank scriptbase straight C subscript 1 space end scriptbase presubscript blank presuperscript space n end presuperscript a to the power of n minus 1 space end exponent b space plus space horizontal ellipsis space plus space scriptbase straight C subscript r space end scriptbase presubscript blank presuperscript n a blank to the power of n minus r end exponent space b blank to the power of r space end exponent plus space horizontal ellipsis space plus space b to the power of n

    • where  scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n equals fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction
      • See below for more information on  scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n
      • You may also see scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n  written as open parentheses n
r close parentheses
or straight C presubscript space n end presubscript subscript r
  • You will usually be asked to find the first three or four terms of an expansion
  • Look out for whether you should give your answer in ascending or descending powers of x
    • For ascending powers start with the constant term, an
    • For descending powers start with the term with x in
      • You may wish to swap a and b over so that you can follow the general formula given in the formula book
  • If you are not writing the full expansion you can either
    • show that the sequence continues by putting an ellipsis (…) after your final term
    • or show that the terms you have found are an approximation of the full sequence by using the sign for approximately equals to (≈)

How do I find the coefficient of a single term?

  • Most of the time you will be asked to find the coefficient of a term, rather than carry out the whole expansion
  • Use the formula for the general term

begin mathsize 22px style scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n space a to the power of n minus r end exponent space b to the power of r end style

  • The question will give you the power of x of the term you are looking for
    • Use this to choose which value of r  you will need to use in the formula
    • This will depend on where the x is in the bracket
    • The laws of indices can help you decide which value of r  to use:
      • For left parenthesis a space plus space b x right parenthesis to the power of n to find the coefficient of x to the power of r use a to the power of n minus r end exponent space left parenthesis b x right parenthesis to the power of r
      • For left parenthesis a space plus space b x squared right parenthesis to the power of n to find the coefficient of x to the power of r use  a to the power of fraction numerator n space minus space r over denominator 2 end fraction end exponent left parenthesis b x squared right parenthesis to the power of r over 2 end exponent
      • For left parenthesis a space plus space b over x right parenthesis to the power of n look at how the powers will cancel out to decide which value of r to use
      • So for open parentheses 3 x blank plus 2 over x close parentheses to the power of 8 to find the coefficient of x squared use the term with r space equals space 3 and to find the constant term use the term with r space equals space 4
      • There are a lot of variations of this so it is usually easier to see this by inspection of the exponents
  • You may also be given the coefficient of a particular term and asked to find an unknown in the brackets
    • Use the laws of indices to choose the correct term and then use the binomial theorem formula to form and solve and equation

Exam Tip

  • Binomial expansion questions can get messy, use separate lines to keep your working clear and always put terms in brackets

Worked Example

Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 3 minus 2 x right parenthesis to the power of 5.1-5-1-binomial-theorem-we-solution-1

The Binomial Coefficient nCr

What is scriptbase bold C subscript bold r end scriptbase presubscript blank presuperscript bold n?

  • If we want to find the number of ways to choose r items out of n different objects we can use the formula for scriptbase straight C subscript r space end scriptbase presubscript blank presuperscript n
    • The formula for r combinations of n items is scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n equals blank fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction
    • This formula is given in the formula booklet along with the formula for the binomial theorem
    • The function scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n can be written open parentheses n
r close parentheses or blank subscript straight n straight C subscript r and is often read as ‘n choose r’
      • Make sure you can find and use the button on your GDC

 

How does scriptbase bold C subscript bold r end scriptbase presubscript blank presuperscript bold n  relate to the binomial theorem?

  • The formula  scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n equals blank fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction is also known as a binomial coefficient
  • For a binomial expansion left parenthesis a space plus space b right parenthesis to the power of n the coefficients of each term will be scriptbase straight C subscript 0 end scriptbase presubscript blank presuperscript n, scriptbase straight C subscript 1 end scriptbase presubscript blank presuperscript n and so on up to scriptbase straight C subscript n end scriptbase presubscript blank presuperscript n
    • The coefficient of the r to the power of t h end exponent term will be scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n
  • scriptbase straight C subscript n space equals space scriptbase straight C subscript 0 end scriptbase presubscript blank presuperscript straight n space equals space 1 end scriptbase presubscript blank presuperscript n
  • The binomial coefficients are symmetrical, so scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n space equals space scriptbase straight C subscript n space minus space r end subscript end scriptbase presubscript blank presuperscript n
    • This can be seen by considering the formula for scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n
    • scriptbase straight C subscript n minus r end subscript end scriptbase presubscript blank presuperscript n equals blank fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial open parentheses n minus left parenthesis n minus r right parenthesis close parentheses factorial end fraction equals blank fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction equals blank n straight C subscript r

Exam Tip

  • You will most likely need to use the formula for nCr at some point in your exam
    • Practice using it and don't always rely on your GDC 
    • Make sure you can find it easily in the formula booklet

Worked Example

Without using a calculator, find the coefficient of the term in x cubed in the expansion of left parenthesis 1 space plus space x right parenthesis to the power of 9.

1-5-1-binomial-coefficient-we-solution-2

Pascal's Triangle

What is Pascal’s Triangle?

  • Pascal’s triangle is a way of arranging the binomial coefficients and neatly shows how they are formed
    • Each term is formed by adding the two terms above it
    • The first row has just the number 1
    • Each row begins and ends with a number 1
    • From the third row the terms in between the 1s are the sum of the two terms above it

 

4.1.1-Binomial-Expansion-Notes-Diagram-3-1024x868

How does Pascal’s Triangle relate to the binomial theorem?

  • Pascal’s triangle is an alternative way of finding the binomial coefficients, scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n
    • It can be useful for finding for smaller values of n without a calculator
    • However for larger values of n it is slow and prone to arithmetic errors
  • Taking the first row as zero, stretchy left parenthesis scriptbase straight C subscript 0 end scriptbase presubscript blank presuperscript space 0 end presuperscript space equals space 1 right parenthesis, each row corresponds to the n to the power of t h end exponent row and the term within that row corresponds to the r to the power of t h end exponent term

Exam Tip

  • In the non-calculator exam Pascal's triangle can be helpful if you need to get the coefficients of an expansion quickly, provided the value of n is not too big 

Worked Example

Write out the 7th row of Pascal’s triangle and use it to find the value of  scriptbase straight C subscript 4 end scriptbase presubscript blank presuperscript 6 space end presuperscript.

1-5-1-pascals-triangle-we-solution-3



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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