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If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b)

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Difficulty: 505-555 Levelx   Algebrax                     
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Schnauss
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink] New post   21 Jul 2016, 17:53
EMPOWERgmatRichC wrote:
Hi erikvm,

This is a "layered" concept and it's easy to get "lost" in this prompt because you're used to solving for the final values in most Quant questions.

Here, the 3 "final" numbers are (X - A), (X - B) and (X - C), but the question is NOT asking for any of the 3 final numbers...it's asking for a "piece" of one of them....the value of B.

To answer it, you have to ignore the A, B and C for a moment and go back to the prior "term"

X^3 - X

This can be factored down into 3 pieces. Here's how...

X^3 - X

First, factor out an X...

(X)(X^2 - 1)

Next, reverse-FOIL the other term....
(X)(X+1)(X-1)

Since we're multiplying 3 terms, it doesn't matter what the order is. I'm going to put them in order from least to greatest...

(X-1)(X)(X+1)

Now, looking at THIS, you can figure out what A, B and C are. Since A>B>C, then....

A = +1
B = 0
C = -1

Final Answer:
Show Spoiler
C


GMAT assassins aren't born, they're made,
Rich



What I found confusing about this problem is that we are not told that the left side of the equation is equal to 0 (x^3-x=0), so how do we know to begin factoring x^3-x here into x(x-1)(x=1)? Am I missing something really basic here?

Thanks in advance!
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EMPOWERgmatRichC
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink] New post   24 Dec 2016, 10:16
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Hi Schnauss,

The 'act' of factoring does not require an equation - it's just a way of re-writing information that you've been given.

For example, if you're told that you have 'two boxes that each contain the same number of widgets', then you can write that as (2)(W)... even though you don't have an actual equation yet. If I wanted to, I could rewrite 2W as (W + W). Certain GMAT questions just come down to organizing information in a way that makes it easy to answer the given question, so you should think about how you choose to take notes and 'translate' sentences ('your way' might not be the only way, and there might be additional 'steps' that you can take to simplify what you have written).

GMAT assassins aren't born, they're made,
Rich
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Sa800
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink] New post   25 Jun 2023, 22:30
my question is why or where are people getting -1,0,1 from? I assume they factor …. but then Why do people automatically set the quadratic equation equal to zero after factoring?

nowhere on the question does it say that….(x)(x+1)(x-1) = 0…..

is there a fundamental theorem im missing?
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink] New post   26 Jun 2023, 00:12
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sa800 wrote:
my question is why or where are people getting -1,0,1 from? I assume they factor …. but then Why do people automatically set the quadratic equation equal to zero after factoring?

nowhere on the question does it say that….(x)(x+1)(x-1) = 0…..

is there a fundamental theorem im missing?


You are missing the crucial piece of information given in the stem: \(x^3-x=(x-a)(x-b)(x-c)\) for ALL numbers \(x\).

Since \(x^3-x=(x-a)(x-b)(x-c)\) is true for ALL \(x-es\) than it must also be true for \(x=0\), \(x=1\) and \(x=-1\). Setting x to 0 1 and -1 allows us to determine the values of a, b, and c.

Hope it helps.
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If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink] New post   12 Mar 2024, 11:23
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Attached is a visual that should help.­  First, factor out x—then use your quadratic identities + logic to solve.  

You must also be familiar with the specific quadratic identity (x+y)(x-y) = x^2 - y^2, or in this case (x+1)(x-1) = x^2 - 1.

­
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If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b) [#permalink]
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