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For how many integer values of \(k\) is \(k^4 + 2k < 2k^3 + k^2\)?
A. 0
B. 1
C. 2
D. 4
E. More than 4
A. 0
B. 1
C. 2
D. 4
E. More than 4
12 Jan 2026, 10:03
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Official Solution:
For how many integer values of \(k\) is \(k^4 + 2k < 2k^3 + k^2\)?
A. 0
B. 1
C. 2
D. 4
E. More than 4
\(k^4 + 2k < 2k^3 + k^2\)
\(k^4 - 2k^3 - k^2 + 2k < 0\)
\(k(k^3 - 2k^2 - k + 2) < 0\)
\(k(k^2(k - 2) - (k - 2)) < 0\)
\(k(k - 2)(k^2 - 1) < 0\)
\(k(k - 2)(k - 1)(k + 1) < 0\)
The transition points are \(k = -1\), \(k = 0\), \(k = 1\), and \(k = 2\) (these are the values of \(k\) at which a factor changes sign). This gives us five ranges:
\(k < -1\)
\(-1 < k < 0\)
\(0 < k < 1\)
\(1 < k < 2\)
\(k > 2\)
Next, test an extreme value for \(k\): if \(k\) is a large enough number, say 100, then all factors will be positive, resulting in a positive value for the whole expression. Therefore, when \(k > 2\), the expression is positive. Now, here’s the trick: since the expression is positive in the 5th range, it will be negative in the 4th range, positive again in the 3rd range, negative in the 2nd range, and positive in the 1st range, following the pattern: +, -, +, -, +.
Thus, the expression is negative for \(-1 < k < 0\) and \(1 < k < 2\).
Since \(k\) must be an integer, there are no integer values satisfying the inequality.
Answer: A
For how many integer values of \(k\) is \(k^4 + 2k < 2k^3 + k^2\)?
A. 0
B. 1
C. 2
D. 4
E. More than 4
\(k^4 + 2k < 2k^3 + k^2\)
\(k^4 - 2k^3 - k^2 + 2k < 0\)
\(k(k^3 - 2k^2 - k + 2) < 0\)
\(k(k^2(k - 2) - (k - 2)) < 0\)
\(k(k - 2)(k^2 - 1) < 0\)
\(k(k - 2)(k - 1)(k + 1) < 0\)
The transition points are \(k = -1\), \(k = 0\), \(k = 1\), and \(k = 2\) (these are the values of \(k\) at which a factor changes sign). This gives us five ranges:
\(k < -1\)
\(-1 < k < 0\)
\(0 < k < 1\)
\(1 < k < 2\)
\(k > 2\)
Next, test an extreme value for \(k\): if \(k\) is a large enough number, say 100, then all factors will be positive, resulting in a positive value for the whole expression. Therefore, when \(k > 2\), the expression is positive. Now, here’s the trick: since the expression is positive in the 5th range, it will be negative in the 4th range, positive again in the 3rd range, negative in the 2nd range, and positive in the 1st range, following the pattern: +, -, +, -, +.
Thus, the expression is negative for \(-1 < k < 0\) and \(1 < k < 2\).
Since \(k\) must be an integer, there are no integer values satisfying the inequality.
Answer: A
