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General Binomial Expansion (CIE A Level Maths: Pure 3)
Revision Note
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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