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Old 5th January 2012, 09:24 PM   #1 (permalink)
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Quadratic Expansion using Binomial Theorem

Hi, I am stuck with a question about Binomial Expansion using Binomial Theorem, here is the question:

Find, in ascending powers of x, the first 4 terms in the expansion of (1 - 2x - 4x^2)^11

the solution is 1 - 22x + 176x^2 - 440x^3 + .......

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Old 6th January 2012, 12:49 AM   #2 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

Many ways to approach this problem, the usual designed way is to factorize 1 - 2x - 4x^2 = 4[(sqrt(5) - 1)/4 - x][x + (sqrt(5) + 1)/4] then from here you should know how to expand, but the factorization is a bit weird, so maybe we can treat 2x + 4x^2 as a single term and expand, we'll have
2x + 4x^2 = u
(1 - 2x - 4x^2)^11
= (1 - u)^11
= (11C0) - (11C1)u + ...
= (11C0) - (11C1)(2x + 4x^2) + (11C2)(2x + 4x^2)^2 - (11C3)(2x + 4x^2)^3 + ...

Note here, we'll stop at this term because (2x + 4x^2)^3 is the last term which will yield an x^3 in it, which is the last term we want. Also, don't use 1 - 2x = u because you will then have 2 terms that have x in it which will cause unnecessary complications.

(1 - 2x - 4x^2)^11
= (11C0) - (11C1)(2x + 4x^2) + (11C2)(2x + 4x^2)^2 - (11C3)(2x + 4x^2)^3 + ...
= 1 - 11(2x + 4x^2) + 55(4x^2 + 16x^3 + ... ) - 165(8x^3 + ...) + ...
= 1 - 22x - 44x^2 + 220x^2 + 880x^3 - 1320x^3 +...
= 1 - 22x + 176x^2 - 440x^3 + ...

lol i guess this way is correct, also notice that i expand (2x + 4x^2)^2 = (4x^2 + 16x^3 + ... ) and (2x + 4x^2)^3 = 165(8x^3 + ...) because any further we'll get a degree higher than 3, which is unneeded. Anyway, you don't have to substitute 2x + 4x^2 = u in your homework or whatever, i just did this to show you that i am just treating 2x + 4x^2 as one single term.


Last edited by Betakuwe; 6th January 2012 at 12:55 AM.
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Old 7th January 2012, 11:56 AM   #3 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

how did you get 2x + 4x^2

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Old 7th January 2012, 10:20 PM   #4 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

Originally Posted by danny2010 View Post
how did you get 2x + 4x^2
He got the 2x + 4x^2 from:
(1 - 2x - 4x^2) = (1 - [2x + 4x^2])

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Old 7th January 2012, 11:12 PM   #5 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

Originally Posted by danny2010 View Post
how did you get 2x + 4x^2
aiyo this kind of small thing also need ask de meh? Shouldn't expect me to feed you everything, you yourself should also think a bit. megaman have answered the question anyway.

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Old 8th January 2012, 09:10 AM   #6 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

Originally Posted by Betakuwe View Post
aiyo this kind of small thing also need ask de meh? Shouldn't expect me to feed you everything, you yourself should also think a bit.
I was just asking only alright to make sure of it, no harm. Even if it is very simple to you, do you think it is simple to me? now this reply of yours pissed me off okay? btw thanks for your help I don't mean to be rude, if I know how to solve this question, I wouldn't have waste my time asking here and you would not waste your time helping me to solve this question correct or not?

Originally Posted by megaman123 View Post
He got the 2x + 4x^2 from:
(1 - 2x - 4x^2) = (1 - [2x + 4x^2])
I see, I would thought I need to factorize -2x out since -2x - 4x^2 the common factor is -2x lol so he factorize out the minus sign only. Quite tricky thanks for your point out


Last edited by danny2010; 8th January 2012 at 09:27 AM.
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Old 8th January 2012, 10:52 AM   #7 (permalink)
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Re: Quadratic Expansion using Binomial Theorem

Originally Posted by danny2010 View Post
I was just asking only alright to make sure of it, no harm. Even if it is very simple to you, do you think it is simple to me? now this reply of yours pissed me off okay? btw thanks for your help I don't mean to be rude, if I know how to solve this question, I wouldn't have waste my time asking here and you would not waste your time helping me to solve this question correct or not?


I see, I would thought I need to factorize -2x out since -2x - 4x^2 the common factor is -2x lol so he factorize out the minus sign only. Quite tricky thanks for your point out
oh sorry then. Anyway you need not opt for 1 - 2x - 4x^2 = 1 - (2x + 4x^2), you can also go with 1 - 2x - 4x^2 = 1 + (-2x - 4x^2), either way it will yield you the same result, just to remind you that maths is a flexible subject and there is not rigid correct or wrong, as long as it makes sense, you're good.

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