General Binomial Expansion (Cambridge (CIE) A Level Maths) : Revision Note
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General Binomial Expansion
What is the general binomial expansion?
The binomial expansion applies for positive integers, n ∈ ℕ
The general binomial expansion applies to other types of powers too

The general binomial expansion applies for all real numbers, n ∈ℝ
Usually fractional and/or negative values of n are used
It is derived from (a + b)n, with a = 1 and b = x
a = 1 is the main reason the expansion can be reduced so much

Unless n ∈ ℕ, the expansion is infinitely long
It is only valid for |x| < 1
This is another way of writing -1 < x < 1
This is often called the validity statement
The restriction |x| < 1 means the series will converge
Higher powers of x can be ignored (as r → ∞, xr → 0)
Only the first few terms of an expansion are needed
How do I expand brackets with the general binomial expansion?
STEP 1 Write the expression in the form (1 + x)n
STEP 2 Expand and simplify
Use a line for each term to make things easier to read and follow
Use brackets - fractions and negatives get ugly!
STEP 3 If required, check and state the validity statement

How do I use the general binomial expansion when it is (1 + bx)?
STEP 1 Write the expression in the form (1 + bx)n
STEP 2 Replace “x” by “bx” in the expansion
Check carefully to see if b is negative
STEP 3 Expand and simplify
Use a line for each term to make things easier to read and follow
Use brackets
STEP 4 If required, check and state the validity statement The validity statement changes
Replace “x” with “bx” so now |bx| < 1


Worked Example


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