What is the general binomial expansion?

• The binomial expansion applies for positive integers,?n?∈??
• The?general?binomial?expansion?applies to other types of powers too

• The?general?binomial?expansion?applies for all?real?numbers,?n?∈?
• 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 for (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