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What is the number of roots of the equation (x^3-1)/(x^2+x+1

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by Max@Math Revolution » Mon Oct 22, 2018 12:24 am

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[Math Revolution GMAT math practice question]

What is the number of roots of the equation (x^3-1)/(x^2+x+1)=x?

A. 1
B. 2
C. 3
D. 4
E. no solution
Last edited by Max@Math Revolution on Thu Oct 25, 2018 10:44 pm, edited 1 time in total.
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by fskilnik@GMATH » Mon Oct 22, 2018 4:17 am

What is the number of roots of the equation (x^3-1)/(x^2+x+1)=x?

A. 4
B. 3
C. 2
D. 1
E. 0
$$?\,\,\,:\,\,\,\# \,\,{\rm{roots}}\,\,{\rm{of}}\,\,\,\,{{{x^{\rm{3}}} - 1} \over {{x^2} + x + 1}} = x\,\,\,\,\left( * \right)$$
$${x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{{{x^3} - 1} \over {{x^2} + x + 1}} = x - 1\,\,\,\,\,\,\,\,\,\left[ {\,{x^2} + x + 1 \ne 0\,\,\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,} \right]$$
$$\,\left( * \right)\,\,\, \Rightarrow \,\,\,\,x - 1 = x\,\,\,,\,\,\,\,{\rm{no}}\,\,{\rm{roots}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,?\,\,\,:\,\,\,{\rm{0}}\,\,{\rm{roots}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left( E \right)$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Wed Oct 24, 2018 12:08 am
=>

The original condition (x^3-1)/(x^2+x+1)=x is equivalent to x - 1 = x as shown below:

(x^3-1)/(x^2+x+1)=x
=> (x-1) (x^2+x+1) / (x^2+x+1) = x by factoring
=> x - 1 = x

But x - 1 = x has no solution.

Therefore, the answer is E.
Answer: E
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