Binomial Expansion (P1.8 & 2.4)

Question 1

describe Pascal’s triangle

Answer 1

adding adjacent pairs of numbers –> the numbers on the next row

the (n+1)th row of Pascal’s triangle gives the coefficients in the expansion of (a+b)^n

Question 2

how do you find values of Pascal’s triangle on calculator?

Answer 2

nCr or (n r)
nCr = n! / r!(n-r)!
the rth entry in the nth row of Pascal’s triangle is given by (n-1)C(r-1) = (n-1 r-1)

Question 3

binomial expansion with integer n values

Answer 3

see Wilson OneNote

Question 4

why can binomial expansion be used to find simple approximations for complicated functions?

Answer 4

if x<1 x^n gets smaller as n increases
converges to an answer as # of terms of the expansion increases
if x is small large powers can be ignored to approximate function/estimate a value

Question 5

when is binomial expansion valid?

Answer 5

|x|<1
-1 < x < 1

Question 6

when are approximations based on binomial expansion more accurate?

Answer 6

when more terms of the expansion are used
when the values of x substituted into the expansion are closer to 0

Question 7

when is the expansion of (1+bx)^n valid?

Answer 7

when |bx| < 1
|x|< 1/|b|

Question 8

expanding (1+x)^n

Answer 8

use formula book

Question 9

when n is not a natural number…

Answer 9

none of the factors = 0 in formula
so this version gives an infinite # of terms

Question 10

what is the method to expand (a+bx)^n?

Answer 10

a^n (1 + b/a x)^n

Question 11

when is the expansion of (a+bx)^n valid?

Answer 11

when |b/a x| < 1
|x|<|a/b|

Question 12

what is the method for binomial expansion with partial fractions?

Answer 12

1. split into partial fractions
2. work out each expansion separately
3. add or subtract the 2 expansions

Question 13

what is the method for using a substitution of x to find an approximation for a given value?

Answer 13

1. substitute in value for x into the expanded & non-expanded form
2. make the non-expanded form into the given value, doing the same to the expanded side