AS Maths: Bionomial Expansion! Please expand!
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AS Maths: Bionomial Expansion! Please expand!

[From: ] [author: ] [Date: 12-04-28] [Hit: ]
(6C0)*1 + (6C1)*2x + (6C2)*4x² + (6C3)*8x³ +...1 + 12x + 60x² + 160x³ +.........
Write down and siimplify the first four terms in the binomial expansion of (1+2x)^6.

I don't get what to do it to. What is the first four terms?

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The first term (by the order written, and given they haven't stated ascending of descending powers of x) is 1^6 = 1, so
(6C0)*1 + (6C1)*2x + (6C2)*4x² + (6C3)*8x³ +... which is
1 + 12x + 60x² + 160x³ +...

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ok here is what i do for this stuff

write down pascals
(if you dont know it look it up , so)
go to the seventh row down, this is because 1 at the top = 0

ok then write down the numbers leaving a space inbetween (only four since it only asks for four)

Then you are going to do this
from left to right
(1) 1^6 X 2x^0 (6) 1^5 X 2x^1 (15) 1^5 X 2x^2 (you continue this till you have 4 terms)
((z) = pascals number here)
then what you do is multiply it out
so you would get

(1)1X1 = 1
(6)1 X 2x = 12x
(15)1 X 4x^2 = 60x^2 (you have to square the 2 here to get 4)
so you get 1+12x+60x^2 (ill let you finish it off for practice) you need 1 more term

I hoped i helped, if not add more details and ill see what i can do

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1 (1)^6 | (2x)^0 = 1
6 (1)^5 | (2x)^1 = 12x
15 (1)^4 | (2x)^2 = 60x^2
20 (1)^3 | (2x)^3 = 160x^3
15 (1)^2 | (2x)^4 = 240x^4
6 (1)^1 | (2x)^5 = 190x^5
1 (1)^0 | (2x)^6 = 64x^6

so (1 + 2x)^6 = 1 + 12x + 60x^2 + 160x^3 + 240x^4 + 190x^5 + 64x^6

as you asked for the first four terms = 1 + 12x + 60x^2 + 160x^3

Hope i helped you out, this method i have used is the easiest to do.

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(6C0 * 1^6 * (2x)^0) + (6C1 * 1^5 * (2x)^1) + (6C2 * 1^4 * (2x)^2) + (6C3 * 1^3 * (2x)^3) + ... =
(1 * 1 * 1) + (6 * 1 * 2x) + ((6! / (4!2!)) * 1 * 4x^2) + ((6! / (3!3!)) * 1 * 8x^3) + ... =
1 + 12x + 60x^2 + 160x^3 + ...
1
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