4.1.5
Binomial Expansion & Rational Powers 2 (A2 only)
Binomial Expansion for Rational Powers
Binomial Expansion for Rational Powers
We can use the binomial expansion of to expand expressions of the form .
Taking out factors
Taking out factors
- In order to expand using the binomial expansion, we need to take out a factor of
and write it in front of the brackets:
- We can then expand using the binomial expansion equation in the formula book, remembering to replace any terms with .
- Don't forget to multiply every term in the expansion by at the end to give the final answer.
Example
Example
- What are the first four terms of the binomial expansion of ?
- What are the range of values of for which the expansion is valid?
Take out factors
Take out factors
- In order to use the expansion of , we need to take out a factor of
from
"
Expand
Expand
- The first term of the expansion of is equal to 1.
Second term
Second term
- The second term of the binomial expansion of is .
- Substituting in and noticing that we must change
, we get
Third term
Third term
- The third term of the binomial expansion is .
- Substituting in and changing
, we get:
Fourth term
Fourth term
- The third term of the binomial expansion is .
- Substituting in and changing
, we get:
Add terms
Add terms
- The expansion of is equal to:
Multiply each term by
Multiply each term by
- To find the final answer, we need to multiply the whole expansion by :
Validity
Validity
- The binomial expansion is valid when , which means:
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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