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Which of the following is a value of x for which X^11 - x^3

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Director
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Joined: 06 Sep 2006
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Which of the following is a value of x for which X^11 - x^3 [#permalink]

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New post 08 Apr 2007, 09:39
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A
B
C
D
E

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Which of the following is a value of x for which X^11 - x^3 > 0?

A. -2
B. -1
C. -1/2
D. 1/2
E. 1
Manager
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Re: PS - Algebra [#permalink]

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New post 08 Apr 2007, 10:27
asaf wrote:
Which of the following is a value of x for which X^11 - x^3 > 0?

A. -2
B. -1
C. -1/2
D. 1/2
E. 1


Answer D


Above can be simplified to
x^3(x^8 - 1)
= x^3 (x^4 + 1)(x^2 + 1)(X + 1)(x - 1)

if this is greater than 0 then x cant be 1, -1. If x is 1/2 then x^11 < x ^3 so its less than 0 (higher power of positive fractions are less than lower ones)

-2 is also out as (-2) ^ 11 is a BIG negative

Whats left is -1/2
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 [#permalink]

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New post 08 Apr 2007, 10:36
(C) for me :)

We could answer pretty fast by observing the behaviour of x^11 and x^3 in the XY Plan.

Basically, the question asks x^11 > x^3 that means for which value of x do we have the draw of x^11 "above" the draw of x^3.

Drawing the 2 functions, we obtain the fig 1. By this way, we can see that it's for x=-1/2.
Attachments

Fig1_X power 11 and X power 3.gif
Fig1_X power 11 and X power 3.gif [ 3.86 KiB | Viewed 1346 times ]

Senior Manager
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 [#permalink]

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New post 08 Apr 2007, 10:59
x^3(x^8-1)>0 ?
A. negative * positive <0> 0
D. positive * negative <0
E. 1 (1-1) = 0
Answer is C.
Director
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 [#permalink]

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New post 08 Apr 2007, 17:55
10 second way to answer this question:

factor out x^3, and you get x^3(x^8-1)

x^8 is positive since the exponent is even, first exponent is odd so first term will be negative

so, try -1/2 quickly, (-1/2)^3((-1/2)^8-1) = negative(<0>0

Done

Practice manipulating equations in your head.
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 [#permalink]

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New post 08 Apr 2007, 19:24
x^3(X^8 - 1) > 0

Only reasonable value is -1/2. This will result in x^3 giving you a negative and (x^8-1) giving you a negative. Multiply both to get a positive.


The remaining choices either give a negative product or a product of 0.

Ans C
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Re: PS - Algebra [#permalink]

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New post 15 Jun 2009, 14:25
I dont think you need any calculations at all for this one, not even the factoring. You can immediately get rid of B and E since 1^anything is 1 and 1-1 = 0.

You can also get rid of A since the exponents are odd.

Most people know that values between between 1 and 0 get smaller the larger the exponent. Just in case you might wanna try to plug in C and D though.
Re: PS - Algebra   [#permalink] 15 Jun 2009, 14:25
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Which of the following is a value of x for which X^11 - x^3

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