4.1.4
Binomial Expansion & Rational Powers (A2 only)
Binomial Expansion with Rational Powers
Binomial Expansion with Rational Powers
If we have an expression and is a negative or rational number, we need to use a different equation for its expansion.
Binomial expansion
Binomial expansion
- The binomial expansion of a the expression is equal to:
- This is an infinite sum of terms, also known as an infinite series.
- This expansion is valid when and ℝ (is a real number).
- This equation is included in the formula booklet in the exam.
Binomial expansion
Binomial expansion
- We can use this equation to work out the expansion of binomials of the form by grouping together
and treating it as if it was
:
- This expansion is valid when or .
Example
Example
- What are the first four terms of the binomial expansion of ?
- For which values of is the expansion valid?
Comparing variables
Comparing variables
- If we compare the expression we want to expand with the equation given in the formula booklet, we see that .
- So we can substitute this into each term to find the binomial expansion.
First term
First term
- The first term of this type of binomial expansion is always equal to 1.
Second term
Second term
- The second term of the binomial expansion is .
- Substituting in , we get:
Third term
Third term
- The third term of the binomial expansion is .
- Substituting in , we get:
Fourth term
Fourth term
- The third term of the binomial expansion is .
- Substituting in , we get:
Answer
Answer
- The first four terms of the binomial expansion of are:
Validity
Validity
- The expansion is valid for values of such that <1.
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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