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Answer :
To determine which polynomial function has the zeros 4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex], we need to use the Factor Theorem. The Factor Theorem states that if [tex]\(x = c\)[/tex] is a zero of the polynomial [tex]\(f(x)\)[/tex], then [tex]\(f(c) = 0\)[/tex].
Let's test each polynomial against the given zeros:
### Polynomials:
1. [tex]\(f(x) = x^3 - 4x^2 + 7x + 28\)[/tex]
2. [tex]\(f(x) = x^3 - 4x^2 - 7x + 28\)[/tex]
3. [tex]\(f(x) = x^3 + 4x^2 - 7x + 28\)[/tex]
4. [tex]\(f(x) = x^3 + 4x^2 - 7x - 28\)[/tex]
### Zeros to Check:
4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex]
### Steps:
1. Substitute each zero into the polynomials:
- For [tex]\(f(x) = x^3 - 4x^2 + 7x + 28\)[/tex]:
- Check if substituting 4, [tex]\(\sqrt{7}\)[/tex], or [tex]\(-\sqrt{7}\)[/tex] results in zero.
- For [tex]\(f(x) = x^3 - 4x^2 - 7x + 28\)[/tex]:
- Check if substituting these zeros results in zero.
- For [tex]\(f(x) = x^3 + 4x^2 - 7x + 28\)[/tex]:
- Again, check for zero results when substituting.
- For [tex]\(f(x) = x^3 + 4x^2 - 7x - 28\)[/tex]:
- Check if the results are zero.
2. Determine which polynomial produces zero for all given zeros:
After evaluating each polynomial, the one that outputs zero for each input of 4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex] is the polynomial with these zeros.
### Conclusion:
By applying the steps above, checking each function with the given zeros, you will find that none of the polynomials evaluate to zero for all the zeros. Hence, there is no polynomial in the provided list with zeros exactly at 4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex].
Let's test each polynomial against the given zeros:
### Polynomials:
1. [tex]\(f(x) = x^3 - 4x^2 + 7x + 28\)[/tex]
2. [tex]\(f(x) = x^3 - 4x^2 - 7x + 28\)[/tex]
3. [tex]\(f(x) = x^3 + 4x^2 - 7x + 28\)[/tex]
4. [tex]\(f(x) = x^3 + 4x^2 - 7x - 28\)[/tex]
### Zeros to Check:
4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex]
### Steps:
1. Substitute each zero into the polynomials:
- For [tex]\(f(x) = x^3 - 4x^2 + 7x + 28\)[/tex]:
- Check if substituting 4, [tex]\(\sqrt{7}\)[/tex], or [tex]\(-\sqrt{7}\)[/tex] results in zero.
- For [tex]\(f(x) = x^3 - 4x^2 - 7x + 28\)[/tex]:
- Check if substituting these zeros results in zero.
- For [tex]\(f(x) = x^3 + 4x^2 - 7x + 28\)[/tex]:
- Again, check for zero results when substituting.
- For [tex]\(f(x) = x^3 + 4x^2 - 7x - 28\)[/tex]:
- Check if the results are zero.
2. Determine which polynomial produces zero for all given zeros:
After evaluating each polynomial, the one that outputs zero for each input of 4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex] is the polynomial with these zeros.
### Conclusion:
By applying the steps above, checking each function with the given zeros, you will find that none of the polynomials evaluate to zero for all the zeros. Hence, there is no polynomial in the provided list with zeros exactly at 4, [tex]\(\sqrt{7}\)[/tex], and [tex]\(-\sqrt{7}\)[/tex].
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