Final answer:
The integer bounds of the real zeros for the polynomial f(x)=x^3-4x^2+25x-100 can be identified by the Upper and Lower Bounds of Zeros theorem as -4 and 4 respectively.
Step-by-step explanation:
The question asks us to use synthetic division to identify integer bounds of the real zeros of the polynomial f(x)=x^3−4x^2+25x−100. According to the Descartes' Rule of Signs, there are no positive zeros because f(x) has no sign changes and up to 3 negative zeros because f(-x) has three sign changes. To determine the specific bounds using the Upper and Lower Bounds of Zeros theorem, we need to test integer values of x by dividing the polynomial by (x - k) using synthetic division.
Starting with the possible upper bound, we can test positive integer values. Since the coefficients for the polynomial do not change sign from positive to negative or vice versa, the Upper Bound is determined by the largest coefficient in absolute value which is 25. However, we test integers up to 5 to find the least upper bound which is a coefficient 4. For the lower bound, we do the same with negative integers, and we will find that the least lower bound is determined by the smallest coefficient in absolute value which is -4. Therefore, the correct bounds are as follows:
- Upper bound: 4
- Lower bound: -4