If P(x) is a polynomial with P(–6) = –1.12 and P(–5) = 0.6 then which of the following must be true?
A. There is a zero for P(x) between x = –6 andx = –5.
B. There is a zero for P(x) between x = –1.12 and x = 0.6.
C. There is a zero for P(x) between x = –1.12 and x = 0.6, but it may be a complex number.
D. There is a zero for P(x) between x = –6 and x = –5, but it may be a complex number.
To determine which of the options must be true, we need to consider the given information and the properties of polynomials.
The fact that P(-6) = -1.12 and P(-5) = 0.6 tells us that the polynomial P(x) has a change in sign between x = -6 and x = -5. This means that the polynomial crosses the x-axis somewhere between these two values.
Based on this information, we can conclude that option A must be true: There is a zero for P(x) between x = -6 and x = -5. This is because the polynomial changes sign, indicating that it must have a root (zero) in between these values.
As for the other options:
Option B - There is a zero for P(x) between x = -1.12 and x = 0.6: This option cannot be determined from the given information. The values -1.12 and 0.6 are simply the function values of the polynomial at specific points and do not provide enough information about the position of the zeros.
Option C - There is a zero for P(x) between x = -1.12 and x = 0.6, but it may be a complex number: Again, the given information does not provide enough details about the location of the zeros. Complex zeros are a possibility, but cannot be determined based on the given data.
Option D - There is a zero for P(x) between x = -6 and x = -5, but it may be a complex number: Like option C, complex zeros are a possibility, but the information given does not allow us to determine if they exist or not.
In summary, the only statement that can be determined to be true based on the given information is option A: There is a zero for P(x) between x = -6 and x = -5.