Question

How do i do 3 part a ?

Answer 1

In general the binomial expansion is

`(a+b)^n = {n \choose 0} a^0 b^n + {n \choose 1} a^1 b^(n-1) + {n \choose 2} a^2 b^(n-2) + ... + {n \choose n} a^n b^0`

So in our case, because we want ascending powers of x we'll write,

`(-3x + 1)^(11) = {11 \choose 0} (-3x)^0 1^(11) + {11 \choose 1} (-3x)^(1) 1^(10) + {11 \choose 2} (-3x)^(2) 1^9 + {11 \choose 3 } (-3x)^3 1^8 + ...`

We need to calculate the binomial coefficients:

`{11 \choose 0} = 1`

`{11 \choose 1} = 11`

`{11 \choose 2} = (11 * 10)/(2) = 55`

`{11 \choose 3} = (11 * 10 * 9)/(3 * 2) = 165`

`(-3x+1)^(11) = 1 (-3x)^0 1^(11) + 11(-3x)^(1) 1^(10) + 55 (-3x)^2 1^(9) + 165 (-3x)^3 1^8 + ...`

`(1-3x)^(11) = 1 -33 x + 495 3x^2 - 4455 x^3+ ...`