Determine the intervals on which the following polynomial is entirely negative and those on which it is entirely positive.
A)entirely negative: ; entirely positive: ,
B)entirely negative: ; entirely positive:
C)entirely negative: , ; entirely positive:
D)entirely negative: ; entirely positive:
E)entirely negative: , ; entirely positive:

Question

Determine the intervals on which the following polynomial is entirely negative and those on which it is entirely positive. 
A)entirely negative: ; entirely positive: , 
B)entirely negative: ; entirely positive: 
C)entirely negative: , ; entirely positive: 
D)entirely negative: ; entirely positive: 
E)entirely negative: , ; entirely positive:

Quizplus · Accepted Answer

The polynomial has a degree of 4 and its leading coefficient is positive. Therefore, as x approaches positive infinity or negative infinity, the polynomial also approaches positive infinity. This means that the polynomial has a global minimum somewhere in the middle. In order for the polynomial to be entirely negative or entirely positive, it must not have any real roots. By using Descartes' rule of signs, we can determine that there are no positive roots and either 2 or 0 negative roots. By testing values, we can find that the polynomial has two distinct real roots. Therefore, the polynomial is entirely negative on the intervals between its roots, and entirely positive outside those intervals. Only choice C satisfies these conditions.

Determine the Intervals on Which the Following Polynomial Is Entirely