Binomial Theorem (II)
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In the expansion of (1 + x)(a - bx)^12, where ab is not equal to 0, the coefficient of x^8 is zero. Find in its simplest form the value of the ratio a/b.
I expanded (a - bx)^ 12 using binomial theorem. So you get
(+....-729a^5b^7x^7 + 495a^4b^8x^8 ...+)
Now you'll have (1 + x)(+....-729a^5b^7x^7 + 495a^4b^8x^8 ...+)
Whe you expand it out, you'll get 495a^4b^8 - 729a^5b^7 = 0. I don't think i made any careless mistakes in the expansion as I did it numerous times, but somehow or rather, I still can't derive the answer. It seems awkward (the equation). Or is there an alternative to solving this question? By the way, you should get 5/8 in the end.
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729 or 792? check again...
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(1+x)(a-bx)^12 = (1+x)(a/b-x)^12 (1/b)^12
For (a/b-x)^12,
coefficient of x^7 = 12C7 (a/b)^5 (-1)^7
coefficient of x^8 = 12C8 (a/b)^4 (-1)^8
Therefore,
Coefficient of x^8 = (1/b)^12 [ 12C7(a/b)^5(-1) + 12C8(a/b)^4 ] = 0
a/b = 12C8 / 12C7 = 5/8
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Solving using your working,
495a^4b^8 - 792a^5b^7 = 0
495a^4b^8 = 792a^5b^7
5/8 * b = a
a/b = 5/8
Uncertain's mistake in line 1 isn't significant though.
(1+x)(a-bx)^12 = (1+x)[b(a/b-x)]^12 = (1+x)(a/b-x)^12 (b)^12
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Originally posted by secretliker:
Solving using your working,
495a^4b^8 - 792a^5b^7 = 0
495a^4b^8 = 792a^5b^7
5/8 * b = a
a/b = 5/8
Uncertain's mistake in line 1 isn't significant though.
(1+x)(a-bx)^12 = (1+x)[b(a/b-x)]^12 = (1+x)(a/b-x)^12 (b)^12
So u think u are damn clever? my mistake? is it a mistake to begin with?I write more for TS to understand the concept. In fact, this qn only needs 3 lines to solve <-- can only be understood if and only if the reader is a math pro.
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Huh? What are you talking about? I think you misunderstood me.
This was what you wrote:
(1+x)(a-bx)^12 = (1+x)(a/b-x)^12 (1/b)^12
The bolded part is wrong. That's all.
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chill man
most impt is TS gets the concept/idea of how to do it -
Originally posted by secretliker:
Huh? What are you talking about? I think you misunderstood me.
This was what you wrote:
The bolded part is wrong. That's all.
Sorry sercetliker. Just misread the lines. Ya it is my mistake. -
