Binomial Expansion

Question 1

what is the general equation for the binomial expansion of (1 + x)^n when n is negative or a fraction

Answer 1

1 + nx + (n(n-1)x^2 / 12) + (n(n-1)(n-2)x^3 / 12*3) + ….

Question 2

how would you explain the expansion

Answer 2

• the one is always there (refer later into flashcards)
• for the fractions just multiply n by the next -ve number
• and multiply the consecutive numbers from 1 for the denominator
• and remember the increasing power of x each time

Question 3

for (p + qx)^n, what is the condition for the binomial expansion to be valid if n is negative

Answer 3

|x| < |p/q|

Question 4

how would you binomially expand 1/(1 + x)^2 up to x^3

Answer 4

• it can be written as (1 + x)^-2
• first term is 1
• second term is -2x as n= -2
• next term is [(-2)(-3)]x^2 / 12
• last term is [(-2)(-3)(-4)]x^3 / 123
• making 1 - 2x + 3x^2 - 4x^3

Question 5

what is the domain of x for which the expansion is valid

Answer 5

• |x| < |p/q|
• p = 1 and q = 1
• so |x| < 1

Question 6

what would be the method for binomially expanding a fraction, such as (1+2x)^3 / (1-x)^2

Answer 6

• write it out as a multiplication, so the denominator bracket would have a -ve power
• binomially expand both up to a necessary point (up to the power it asks for)
• multiply both expansions together and tidy up

Question 7

how would you work out the domain of x when two brackets are binomially expanded

Answer 7

• work out the conditions for both brackets
• then combine them together
• if one of the brackets has a +ve power i has no condition, so the total condition would be by the -ve one

Question 8

what is the form you need to rearrange every (p + qx)^n bracket into if you want to binomially expand it

Answer 8

• p^n(1 + qx/p)^n

- this is why the 1 is always at the beginning lol

Question 9

what would you rewrite 1 / root of (4 - x) into

Answer 9

• (4 - x)^-1/2

- 4^-1/2 * (1 - x/4)^-1/2