Subtleties of the General Binomial Expansion (Edexcel A Level Maths: Pure): Revision Note

Subtleties of the 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

• Even when a ≠ 1 the general binomial expansion can still be used

How do I use the general binomial expansion to expand (a+kx)ⁿ?

The general binomial expansion can be applied to expanding (a + kx)ⁿ

STEP 1        Rewrite the question into (1 + bx)ⁿ form

• Roots are fractional powers

• Denominators are negative powers

• A factor may be needed to make “a = 1”

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, |bx| < 1