[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
What is the number of roots of the equation (x^3-1)/(x^2+x+1
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- Max@Math Revolution
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Last edited by Max@Math Revolution on Thu Oct 25, 2018 10:44 pm, edited 1 time in total.
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$$?\,\,\,:\,\,\,\# \,\,{\rm{roots}}\,\,{\rm{of}}\,\,\,\,{{{x^{\rm{3}}} - 1} \over {{x^2} + x + 1}} = x\,\,\,\,\left( * \right)$$
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
$${x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{{{x^3} - 1} \over {{x^2} + x + 1}} = x - 1\,\,\,\,\,\,\,\,\,\left[ {\,{x^2} + x + 1 \ne 0\,\,\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,} \right]$$
$$\,\left( * \right)\,\,\, \Rightarrow \,\,\,\,x - 1 = x\,\,\,,\,\,\,\,{\rm{no}}\,\,{\rm{roots}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,?\,\,\,:\,\,\,{\rm{0}}\,\,{\rm{roots}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left( E \right)$$
This solution follows the notations and rationale taught in the GMATH method.
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Fabio.
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- Max@Math Revolution
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=>
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
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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