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

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If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 19 Feb 2014, 00:19
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b)(x - c) for all numbers x, what is the value of b ?

(A) - 3
(B) -1
(C) 0
(D) 1
(E) 3

Problem Solving
Question: 103
Category: Algebra Simplifying algebraic expressions
Page: 75
Difficulty: 650


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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 19 Feb 2014, 00:19
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SOLUTION

If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b)(x - c) for all numbers x, what is the value of b ?

(A) - 3
(B) -1
(C) 0
(D) 1
(E) 3

As x^3-x=(x-a)(x-b)(x-c) is true for ALL x-es than it must be true for x=0, x=1 and x=-1 too:

x=0 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=-abc, so one of the unknowns equals to zero;

x=1 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=(1-a)(1-b)(1-c), so one of the unknowns equals to 1;

x=-1 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=(-1-a)(-1-b)(-1-c), so one of the unknowns equals to -1.

As a > b > c then a=1, b=0 and c=-1.

Answer: C.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 19 Feb 2014, 01:54
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Option C:
x^3-x can be re-written as:(x-1)*x*(x+1) in ascending order.
(x-1)*x*(x+1)=(x-a)*(x-b)*(x-c)
[As a>b>c=>-a<-b<-c => x-a<x-b<x-c]
Therefore,a=1
b=0
c=-1
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 22 Feb 2014, 05:03
Expert's post
SOLUTION

If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b)(x - c) for all numbers x, what is the value of b ?

(A) - 3
(B) -1
(C) 0
(D) 1
(E) 3

As x^3-x=(x-a)(x-b)(x-c) is true for ALL x-es than it must be true for x=0, x=1 and x=-1 too:

x=0 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=-abc, so one of the unknowns equals to zero;

x=1 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=(1-a)(1-b)(1-c), so one of the unknowns equals to 1;

x=-1 --> x^3-x=(x-a)(x-b)(x-c) becomes: 0=(-1-a)(-1-b)(-1-c), so one of the unknowns equals to -1.

As a > b > c then a=1, b=0 and c=-1.

Answer: C.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 01 May 2014, 07:27
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Alternative SOLUTION
If a, b, and c are constants, a > b > c, and x^3 - x = (x - a)(x - b)(x - c) for all numbers x, what is the value of b ?

(A) - 3
(B) -1
(C) 0
(D) 1
(E) 3


x^3 - x = (x-a)(x-b)(x-c)
x(x^2 - 1) = (x-a)(x-b)(x-c)

Using (a-b)^2 = (a-b)(a+b) algebra form, the expression becomes ( x ) (x - 1)(x + 1) = (x-a)(x-b)(x-c)

Effectively, the solutions are 0, -1, and +1.
As we are given that a > b > c, the order is therefore +1 > 0 > -1
B is therefore 0.

Answer is C
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 17 Jul 2014, 00:07
x^3 - x = (x - a)(x - b)(x - c)

x (x^2 - 1) = (x - a)(x - b)(x - c)

(x + 0) (x+1) (x-1) = (x - a)(x - b)(x - c)

Given that a>b>c; so reorganising LHS of the equation

(x-1) (x + 0) (x+1) = (x - a)(x - b)(x - c)

b = 0

Answer = C
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 17 Jul 2014, 04:17
How do we know so is equal the numbers are -1,0,1. I know there consecutive but why not 2,3,4?
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 17 Jul 2014, 04:43
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bankerboy30 wrote:
How do we know so is equal the numbers are -1,0,1. I know there consecutive but why not 2,3,4?


Please check here: if-a-b-and-c-are-constants-a-b-c-and-x-3-x-x-167671.html#p1335360

As x^3-x=(x-a)(x-b)(x-c) is true for ALL x-es than it must be true for x=0, x=1 and x=-1 too. Substituting these values you get the values of a, b, and c.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x - [#permalink] New post 17 Jul 2014, 17:52
bankerboy30 wrote:
How do we know so is equal the numbers are -1,0,1. I know there consecutive but why not 2,3,4?


Expanding x^3 - x gives the values -1, 0, 1 besides x. Those are derived as shown in my above post

(x+2)(x+3)(x+4) \neq{x^3 - x}
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Re: If a, b, and c are constants, a > b > c, and x^3 - x = (x -   [#permalink] 17 Jul 2014, 17:52
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