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Answer :
Final answer:
To determine which polynomial function has the given zeros using the Factor Theorem, we need to check which functions can be divided evenly by the corresponding factors. By checking each function, we find that the polynomial functions with the zeros 4, √7, and -√7 are f(x) = x³ - 4x² - 7x + 28 and f(x) = x³ + 4x² - 7x - 28.
Explanation:
The Factor Theorem states that if a polynomial function has a zero at a certain value, then the polynomial can be divided evenly by (x - value). Therefore, to determine which of the polynomial functions has the zeros 4, √7, and -√7, we need to check which of the functions can be divided evenly by (x - 4), (x - √7), and (x + √7).
Let's check each function:
- f(x) = x³ - 4x² + 7x + 28
(x - 4) does divide evenly, (x - √7) does not divide evenly, and (x + √7) does not divide evenly. Therefore, this function does not have the zeros 4, √7, and -√7. - f(x) = x³ - 4x² - 7x + 28
(x - 4) does divide evenly, (x - √7) does divide evenly, and (x + √7) does divide evenly. Therefore, this function has the zeros 4, √7, and -√7. - f(x) = x³ + 4x² - 7x + 28
(x - 4) does not divide evenly, (x - √7) does not divide evenly, and (x + √7) does not divide evenly. Therefore, this function does not have the zeros 4, √7, and -√7. - f(x) = x³ + 4x² - 7x - 28
(x - 4) does divide evenly, (x - √7) does divide evenly, and (x + √7) does divide evenly. Therefore, this function has the zeros 4, √7, and -√7.
Based on the analysis above, the polynomial functions that have the zeros 4, √7, and -√7 are f(x) = x³ - 4x² - 7x + 28 and f(x) = x³ + 4x² - 7x - 28.
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