Year 2 chapter 4: Binomial Expansion

Question 1

If ‘n’ from ‘(a+bx)^n’ is a natural number, what formula do you use to find the binomial expansion?

Answer 1

. ‘(a+b)^n = a^n + (nC1)(a^(n-1))(b) + (nC2)(a^(n-2))(b^(2)) + ….. + (nCr)(a^(n-r))(b^(r)) + …. + b^(n)’.

• ‘r’ is a random number that is between 0 and ‘n’.
• There are n+1 terms in this expansion therefore there are a finite number of terms.

Question 2

. If ‘n’ from ‘(1+ x)^n’ is a fraction or/and a negative number, what formula do you use to find the binomial expansion?
. Also explain why this expansion is infinite and why ‘|x|’ or ‘|bx|’ must be less than 1:

Answer 2

‘(1+x)^n = 1 + n(x) + ((n(n-1))/(2!)) (x^(2)) + ((n(n-1)(n-2))/(3!)) (x^(3)) + …. ‘.
. This can only be used for ‘1+bx’, not ‘2+bx’ (for example).
. It is infinite because ‘n’ isn’t a positive integer so (n-1) or (n-2) etc… will never equal 0.
. ‘|x|’ or ‘|bx|’ must be less than 1 so ‘x’ will be smaller (when squared/cubed….) after each expansion, making the APPROXIMATION more accurate.

Question 3

If your finding an approximation of a value, how do you use the binomial expansion to find it when your told to substitute another value into ‘x’?

Answer 3

. Get your value for your binomial expansion (substituting your value into X).
. Also substitute this value into your original un-expanded expression for ‘x’.
. By rearranging (LHS is un-expanded expression and RHS is expanded expression), get the value that you are trying to approximate on it’s own on the LHS.
. Use calculator to calculate what is on RHS and that is your answer.

Question 4

When you have ‘(a+bx)^n’ and the ‘a’ isn’t equal to 1, how do use the binomial expansion to expand this?

Answer 4

. (a+bx)^n = (a((1+(b/a)(x)))^n = (a^n) X (1+(b/a)(x))^n.
. Therefore you can now use the binomial expansion since you now have 1 inside the bracket, at the end of the expansion you just multiply everything by ‘a^n’

Question 5

If you are asked to binomially expand an algebraic fraction that has a quadratic/cubic etc in the denominator, what do you do?

Answer 5

. Factorise the quadratic/cubic etc in the denominator (if not already factorised).
. Express the fraction as partial fractions.
. Binomially expand each partial fraction and add them all up (or subtract, depending on the each partial fraction) to get your answer.