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16 Dec 2016, 01:44
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B
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If the roots of the equation \(x^3\) – a\(x^2\) + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?
(A) - \(\frac{1}{\sqrt{3}}\)
(B) - 1
(C) 0
(D) 1
(E) \(\frac{1}{\sqrt{3}}\)
(A) - \(\frac{1}{\sqrt{3}}\)
(B) - 1
(C) 0
(D) 1
(E) \(\frac{1}{\sqrt{3}}\)
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
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16 Dec 2016, 02:39
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ethanhunt786
If the roots of the equation \(x^3\) – a\(x^2\) + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?
(A) - \(\frac{1}{\sqrt{3}}\)
(B) - 1
(C) 0
(D) 1
(E) \(\frac{1}{\sqrt{3}}\)
(A) - \(\frac{1}{\sqrt{3}}\)
(B) - 1
(C) 0
(D) 1
(E) \(\frac{1}{\sqrt{3}}\)
In order to solve this question we need to know some special formulars for cubic polynomials.
Let's denote our polynomial as:
\(px^3 + rx + qx + s = 0\)
and \(x_1\) , \(x_2\) and \(x_3\) will be the roots of that polynomial.
then:
\(x_1*x_2 + x_1*x_3 + x_2*x_3 = \frac{c}{a}\)
Now back to our question:
our \(p = 1\), \(q=b\)
Roots of our polynomial are consecutive integers =\((y- 1)\) , \(y\) and \((y + 1)\)
and we have:
\(y*(y - 1) + y*(y + 1) + (y - 1)*(y + 1) = b\)
simplifying this we'll get:
\(3*y^2 - 1 = b\) and we need to find min of that function.
Because \(y^2\) is always \(>= 0\), them our b will get min value only when \(y=0\)
Hence \(b min = -1\)
Answer B
Is it really a GMAT question? It seems rather complicated for GMAT.
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THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.
If you would like to discuss this question please re-post it in the respective forum. Thank you!
To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
20 Apr 2022, 23:26
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Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
