Y2, C4 - Binomial Expansion

Question 1

When are infinite binomial expansions valid

Answer 1

-1 < x < 1

Question 2

When is the expansion of (1 - 2/3x)^-1 valid

Answer 2

Modulus of -2/3x = 2/3x
2/3x < 1
x < 3/2

Question 3

When are expansions allowed to be infinite

Answer 3

When they converge

Question 4

What is the n value of root(1-3x)

Answer 4

(1-3x)^-1/2
n = 1/2

Question 5

How do you binomially expand:
sqr(1 / (4 + 5x))

Answer 5

Get in form (1 + bx)^n
4(1 + 5x/4)^-1/2
Binomial expansion on brackets
4 ( 1 + (5x/4)(-1/2) + ((-1/2)(-3/2) / 2!) x (5x)^2

Question 6

Which value gives the best approximation and why?
x = -14
x = -0.5
x = 2

Answer 6

x = -0.5 as it’s modulus gives the smallest value of x

Question 7

How do you binomially expand:
(2x^2 + 5x -10) / (x-1)(x-2)

Answer 7

Write as partial fractions
A + B / (x-1) + C / (x-2)
Algebraic long division gives A as 2
B = -1, C = 4
Expand -1 / (x-1) and 4 / (x-2) separately and add on to the 2 (A)
= 2 + (1 + x + x^2) + (2 - x + x^2 / 2)
Answer = 5 + 3x^2 / 2

Question 8

Why can’t the form -(x - 1)^-1be binomially expanded

Answer 8

Not in (1 + x)^-1
Need to multiply all terms by -1

Question 9

Why can’t the form (x + 2)^-1 be binomially expanded

Answer 9

Not in 2(1 + x/2)^-1
Need to have in form 1 + ax so must divide terms by 2