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then the average of x, x^2, x^3, x^4.

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then the average of x, x^2, x^3, x^4. [#permalink] New post 27 Dec 2015, 18:45
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Question Stats:

78% (01:13) correct 21% (01:30) wrong based on 33 sessions
If \(1+x+x^2+x^3=60\), then the average (arithmetic mean) of \(x\), \(x^2\), \(x^3\), and \(x^4\) is equal to which of the following?

A. 12x
B. 15x
C. 20x
D. 30x
E. 60x


Practice Questions
Question: 9
Page: 157
Difficulty: hard
[Reveal] Spoiler: OA

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Re: then the average of x, x^2, x^3, x^4. [#permalink] New post 27 Dec 2015, 18:55
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Here we have \(1 + x + x^2 + x^3\) = 60.

Now we multiply both sides with x and divide both sides by 4.

\(\frac{x(1 + x + x^2 + x^3)}{4}\) = \(\frac{60*x}{4}\)

Now the LHS corresponds to the average of x, x^2, x^3, x^4. and RHS is 15x.

Hence option B is correct!
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Re: then the average of x, x^2, x^3, x^4. [#permalink] New post 29 Mar 2016, 09:23
How would a student know to multiply by 4 ? Is it intuition? Can you explain the logic/ principle?
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Re: then the average of x, x^2, x^3, x^4. [#permalink] New post 13 Jul 2016, 13:03
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sagnik242 wrote:
How would a student know to multiply by 4 ? Is it intuition? Can you explain the logic/ principle?


The question asks for the arithmetic mean, and that entails dividing by 4 (which is the number of x variables they've given you). Honestly, I was confused by this question too. The process by which we get the answer always make me feel dumb, because it's so obvious when you see it.
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Re: then the average of x, x^2, x^3, x^4. [#permalink] New post 14 Jul 2016, 13:29
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sandy wrote:
If \(1+x+x^2+x^3=60\), then the average (arithmetic mean) of \(x\), \(x^2\), \(x^3\), and \(x^4\) is equal to which of the following?

A. 12x
B. 15x
C. 20x
D. 30x
E. 60x



To find the average of x, x^2, x^3 and x^4, we need to find the SUM of x + x^2 + x^3 + x^4 and divide it by 4.

We're told that 1 + x + x^2 + x^3 = 60, so if we multiply both sides of the equation by x, we get: x(1 + x + x^2 + x^3) = 60x
When we expand the left side, we get: x + x^2 + x^3 + x^4 = 60x

So, the average of x, x^2, x^3 and x^4 = (x + x^2 + x^3 + x^4)/4
= 60x/4
= 15x

Answer:
[Reveal] Spoiler:
B

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Re: then the average of x, x^2, x^3, x^4. [#permalink] New post 23 Jan 2019, 23:29
Expert's post
The first thing I would do is look at the relationship between

x^3+ x^2 + x + 1 and x^4 + x^3 + x^2 + x

if you put them side by side, at least in writing, then you can see that the right expression is simply the left expression times X.

then x^4 + x^3 + x^2 + x= x*(x^3+ x^2 + x + 1) = x*60

(x^4 + x^3 + x^2 + x) = 60x/4= 15x
Re: then the average of x, x^2, x^3, x^4.   [#permalink] 23 Jan 2019, 23:29
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