4.1.3
Binomial Expansion
Binomial Expansion
Binomial Expansion
We can work out the expansion of binomials using an equation involving factorials.
![Illustrative background for Expanding binomials](https://image-v2.cdn.app.senecalearning.com/2020-01/985e38a3-8309-427d-b878-75263ea352a7/algebra-math-write,h_400,q_80,w_640.jpg)
![Illustrative background for Expanding binomials ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/985e38a3-8309-427d-b878-75263ea352a7/algebra-math-write,h_400,q_80,w_640.jpg)
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.
![Illustrative background for Binomial expansion equation](https://image-v2.cdn.app.senecalearning.com/2020-06/8db0d2e2-e041-4cc5-a422-c80cb989be86/Screenshot%202020-06-08%20at%2011.08.37-min,h_400,q_80,w_640.png)
![Illustrative background for Binomial expansion equation ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/8db0d2e2-e041-4cc5-a422-c80cb989be86/Screenshot%202020-06-08%20at%2011.08.37-min,h_400,q_80,w_640.png)
Binomial expansion equation
Binomial expansion equation
- The binomial expansion of is:
- This equation applies when ℕ (is a positive whole number).
- This equation is included in the formula booklet in the exam.
![Illustrative background for General term](https://image-v2.cdn.app.senecalearning.com/2020-06/35bb36cc-8549-427c-ba37-f15992a9eec8/Screenshot%202020-06-08%20at%2011.31.05-min,h_400,q_80,w_640.png)
![Illustrative background for General term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/35bb36cc-8549-427c-ba37-f15992a9eec8/Screenshot%202020-06-08%20at%2011.31.05-min,h_400,q_80,w_640.png)
General term
General term
- The general term in a binomial expansion is given by .
- We can use this to find coefficients of specific orders of variables in the binomial expansion.
![Illustrative background for Example](https://image-v2.cdn.app.senecalearning.com/2020-06/c2960eb3-ad52-4b31-9f94-fc03c557d3bc/Screenshot%202020-06-08%20at%2011.08.39-min,h_400,q_80,w_640.png)
![Illustrative background for Example ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/c2960eb3-ad52-4b31-9f94-fc03c557d3bc/Screenshot%202020-06-08%20at%2011.08.39-min,h_400,q_80,w_640.png)
Example
Example
- Use the binomial theorem to find the expansion of .
![Illustrative background for Comparing variables](https://image-v2.cdn.app.senecalearning.com/2020-06/a8906088-8a0e-400f-820a-6c96ecbc4b5c/Screenshot%202020-06-08%20at%2011.17.11-min,h_400,q_80,w_640.png)
![Illustrative background for Comparing variables ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/a8906088-8a0e-400f-820a-6c96ecbc4b5c/Screenshot%202020-06-08%20at%2011.17.11-min,h_400,q_80,w_640.png)
Comparing variables
Comparing variables
- If we compare the expression we want to expand with the equation given in the formula booklet:
- and .
- So we can substitute these into each term to find the binomial expansion.
![Illustrative background for First term](https://image-v2.cdn.app.senecalearning.com/2020-06/70421774-78de-4d6a-96a1-261170503597/Screenshot%202020-06-08%20at%2011.08.41-min,h_400,q_80,w_640.png)
![Illustrative background for First term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/70421774-78de-4d6a-96a1-261170503597/Screenshot%202020-06-08%20at%2011.08.41-min,h_400,q_80,w_640.png)
First term
First term
- The first term of the binomial expansion is .
- Substituting in and
, we have:
![Illustrative background for Second term](https://image-v2.cdn.app.senecalearning.com/2020-06/a7c77978-4de0-419c-ad76-18208e1f9763/Screenshot%202020-06-08%20at%2011.08.44-min,h_400,q_80,w_640.png)
![Illustrative background for Second term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/a7c77978-4de0-419c-ad76-18208e1f9763/Screenshot%202020-06-08%20at%2011.08.44-min,h_400,q_80,w_640.png)
Second term
Second term
- The second term of the binomial expansion is .
- Substituting in for and
, we have:
- Simplifying this term gives:
![Illustrative background for Third term](https://image-v2.cdn.app.senecalearning.com/2020-06/a7c77978-4de0-419c-ad76-18208e1f9763/Screenshot%202020-06-08%20at%2011.08.44-min,h_400,q_80,w_640.png)
![Illustrative background for Third term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/a7c77978-4de0-419c-ad76-18208e1f9763/Screenshot%202020-06-08%20at%2011.08.44-min,h_400,q_80,w_640.png)
Third term
Third term
- The third term of the binomial expansion is .
- Substituting in for and
, we have:
- Simplifying this term gives:
![Illustrative background for Fourth term](https://image-v2.cdn.app.senecalearning.com/2020-06/9236868a-a7dc-49b3-94ed-d566b26ff656/Screenshot%202020-06-08%20at%2011.08.46-min,h_400,q_80,w_640.png)
![Illustrative background for Fourth term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/9236868a-a7dc-49b3-94ed-d566b26ff656/Screenshot%202020-06-08%20at%2011.08.46-min,h_400,q_80,w_640.png)
Fourth term
Fourth term
- The fourth term of the binomial expansion is .
- Substituting in for and
, we have:
- Simplifying this term gives:
![Illustrative background for Fifth term](https://image-v2.cdn.app.senecalearning.com/2020-06/4a0c38c0-7d59-4d02-9457-b05621a2fc41/Screenshot%202020-06-08%20at%2011.08.47-min,h_400,q_80,w_640.png)
![Illustrative background for Fifth term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/4a0c38c0-7d59-4d02-9457-b05621a2fc41/Screenshot%202020-06-08%20at%2011.08.47-min,h_400,q_80,w_640.png)
Fifth term
Fifth term
- The fifth term of the binomial expansion is .
- Substituting in for and
, we have:
- Simplifying this term gives:
![Illustrative background for Final term](https://image-v2.cdn.app.senecalearning.com/2020-06/7a352d09-d6c4-4bc7-839d-04dd0c2e28b1/Screenshot%202020-06-08%20at%2011.08.51-min,h_400,q_80,w_640.png)
![Illustrative background for Final term ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/7a352d09-d6c4-4bc7-839d-04dd0c2e28b1/Screenshot%202020-06-08%20at%2011.08.51-min,h_400,q_80,w_640.png)
Final term
Final term
- The sixth term of the binomial expansion is .
- We can tell that this is the final term in the expansion because there will be no powers of left in the term and that .
- So this term is simply equal to .
- Substituting in for and
, we have:
![Illustrative background for Sum the terms](https://image-v2.cdn.app.senecalearning.com/2020-06/7a352d09-d6c4-4bc7-839d-04dd0c2e28b1/Screenshot%202020-06-08%20at%2011.08.51-min,h_400,q_80,w_640.png)
![Illustrative background for Sum the terms ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/7a352d09-d6c4-4bc7-839d-04dd0c2e28b1/Screenshot%202020-06-08%20at%2011.08.51-min,h_400,q_80,w_640.png)
Sum the terms
Sum the terms
- The expansion of is equal to the sum of the terms in the binomial expansion, so the answer is:
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
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
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
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
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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