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18 Feb 2020, 02:56
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
61% (03:02) correct
39%
(02:54)
wrong
based on 258
sessions
History
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Not Attempted Yet
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. \(9\)
A. \(5\)
B. \(6\)
C. \(7\)
D. \(8\)
E. \(9\)
18 Feb 2020, 05:03
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MathRevolution
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 see how many distinct values of \(n\) are possible.
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.
20 Feb 2020, 01:56
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=>
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
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
