Problem:
For how many integers x is the number x4−51x2+50 negative?
Answer Choices:
A. 8
B. 10
C. 12
D. 14
E. 16
Solution:
Note that x4−51x2+50=(x2−50)(x2−1), so the roots of the polynomial are ±1 and ±50. Arranged from least to greatest, these roots are approximately −7.1,−1,1,7.1. The polynomial takes negative values on the intervals (−7.1,−1) and (1,7.1), which include (C)12 integers: −7,−6,−5,−4,−3,−2,2,3,4,5,6,7.