To solve this problem, we need to carefully analyze what we know about polynomial functions and zeros. We have been given a specific polynomial function and one of its zeros, and from this information, we need to infer details about other zeros.
Given:
The polynomial function is [tex]\( f(x) = x^3 + 4x^2 + 2x - 1 \)[/tex].
One of its zeros is [tex]\((-1, 0)\)[/tex], which means [tex]\( f(-1) = 0 \)[/tex].
### Step-by-Step Solution:
1. Degree of the Polynomial:
The polynomial [tex]\( f(x) = x^3 + 4x^2 + 2x - 1 \)[/tex] is of degree 3. This means it has a total of 3 zeros (including both real and complex zeros).
2. Given Zero:
It is provided that one of the zeros of this polynomial is [tex]\( x = -1 \)[/tex].
3. Remaining Zeros:
Since the polynomial is of degree 3 and we already know one zero, we need to find the remaining zeros. There must be a total of 3 zeros, and one of them is already known, so there are [tex]\( 3 - 1 = 2 \)[/tex] more zeros left to be found.
4. Nature of Zeros:
- Complex Conjugates: If the polynomial has complex zeros, they must occur in conjugate pairs (because the coefficients of the polynomial are real numbers).
- Real Zeros: We cannot automatically assume all zeros to be real just because one zero is real.
### Possible Nature of the Remaining Zeros:
- The polynomial can have two more real zeros.
- Alternatively, the polynomial can have one real zero and one pair of complex conjugate zeros.
### Conclusions:
- There are two more zeros of the polynomial function [tex]\( f(x) = x^3 + 4x^2 + 2x - 1 \)[/tex].
- Not all the zeros must be real. Some could be complex.
- We have no specific information that guarantees or requires that all zeros must be irrational.
### Correct Statements:
- There are two more zeros.
### Incorrect Statements:
- All the zeros must be real (because some could be complex).
- There are three more zeros (which is incorrect given the degree of the polynomial).
- All the zeros must be irrational (there is no basis to conclude this).
Thus, the correct conclusion based on the given information is: "There are two more zeros."
