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We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,

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Math Revolution GMAT Instructor
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We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,  [#permalink]

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New post 18 Feb 2020, 01:56
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A
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D
E

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  65% (hard)

Question Stats:

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[GMAT math practice question]

We have a polynomial \(x^2 + 4x - n\) for an integer \(n\) between \(1\) and \(100\), inclusive. The polynomial \(x^2 + 4x - n\) is factored into \((x + a)(x + b)\) where \(a\) and \(b\) are integers. How many different values of \(n\) satisfy the polynomial \(x^2 + 4x – n\)?

A. \(5\)

B. \(6\)

C. \(7\)

D. \(8\)

E. \(9\)

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Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,  [#permalink]

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New post 18 Feb 2020, 04:03
MathRevolution wrote:
[GMAT math practice question]

We have a polynomial \(x^2 + 4x - n\) for an integer \(n\) between \(1\) and \(100\), inclusive. The polynomial \(x^2 + 4x - n\) is factored into \((x + a)(x + b)\) where \(a\) and \(b\) are integers. How many different values of \(n\) satisfy the polynomial \(x^2 + 4x – n\)?

A. \(5\)

B. \(6\)

C. \(7\)

D. \(8\)

E. \(9\)


Solution:
    • \(n\) is an integer between \(1\) and \(100\).
    • \(x^2+4x – n\) is factored into \((x+a)(x+b)\)
      o Thus, \(x^2+4x-n = x^2+(a+b) + ab\)
         \(a + b = 4\), and \(ab = -n\)
         Either \(a\) is negative or \(b\) is negative.
    • Let’s consider \(b\) to be negative, because that won’t change the answer because we have to deal the product of \(a\) and \(b\).
    • Let’s see how many distinct values of \(n\) are possible.
Image
For any other value of \(a\) or \(b\), \(n\) will be either less than \(1\) or greater than \(100\), so these are the only possible values of \(n\).
Hence, the distinct possible values of \(n\) are \(8\), the correct answer is Option D.
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Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,  [#permalink]

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New post 20 Feb 2020, 00:56
=>

We can assume \(a < b\) without loss of generality.

Since we have \(a + b = 4\) and \(ab = -n\), we have the following pairs of \((a, b).\)

\((-1, 5), (-2, 6), (-3, 7), (-4, 8), (-5, 9), (-6, 10), (-7, 11), (-8, 12)\)

Then, the possible values of \(n\) are \(5, 12, 21, 32, 45, 60, 77\) and \(96.\)

We have \(8\) possible values of \(n\).

Therefore, the answer is D.
Answer: D
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Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,   [#permalink] 20 Feb 2020, 00:56

We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,

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