Answer:
There is only one sign change in the function f(-x), indicating that there is exactly one negative real zero. The possible number of negative real zeros is one.
I hope this helped answer your question
If you could give me brainliest, it would help out a ton!
Step-by-step explanation:
To determine the number of possible positive and negative real zeros of the function f(x) = 4x^3 - 12x^2 - 5x + 1, we can use the Descartes' Rule of Signs.
1. Counting the number of sign changes in the function:
- Starting from the leftmost term, the sign is positive (+).
- Moving to the next term, the sign changes to negative (-).
- Moving to the next term, the sign changes to positive (+).
- Finally, moving to the last term, the sign remains positive (+).
There are two sign changes in the function, indicating that there are either two positive real zeros or zero positive real zeros.
2. Counting the number of sign changes in the function f(-x):
- Substitute -x for x in the function: f(-x) = 4(-x)^3 - 12(-x)^2 - 5(-x) + 1.
- Simplifying, we get f(-x) = -4x^3 - 12x^2 + 5x + 1.
- Starting from the leftmost term, the sign is negative (-).
- Moving to the next term, the sign remains negative (-).
- Moving to the next term, the sign changes to positive (+).
- Finally, moving to the last term, the sign remains positive (+).
There is only one sign change in the function f(-x), indicating that there is exactly one negative real zero.
Based on the Descartes' Rule of Signs, the possible number of positive real zeros is either two or zero, and the possible number of negative real zeros is one.
