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1. If [tex]$(-1,0)$[/tex] is a zero of the polynomial function [tex]$f(x) = x^3 + 4x^2 + 2x - 1$[/tex], what can you say about the remaining zeros of this function without factoring?

A. There are two more zeros.
B. All the zeros must be real.
C. There are three more zeros.
D. All the zeros must be irrational.

Sagot :

GinnyAnswer
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."
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