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
• a = 1?is the main reason the expansion can be reduced so much

• Unless?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?→?∞,?x^r?→?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