High School

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Using the factor theorem, which of the polynomial functions has the zeros 4, [tex]\sqrt{7}[/tex], and [tex]-\sqrt{7}[/tex]?

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]

Answer :

Final answer:

The polynomial with zeros 4, √7, and -√7 is f(x) = x³ - 4x² + 7x - 28, which corresponds to option 1. This conclusion is derived from the factor theorem and the fact that (x - √7)(x + √7) simplifies to x² - 7. Comparing the given options, we can conclude that the correct polynomial function is 1) f(x) = x³ - 4x² + 7x - 28.

Explanation:

To determine which polynomial function has the zeros 4, √7, and -√7, we can apply the factor theorem. According to the factor theorem, if 'a' is a zero of the polynomial function f(x), then (x - a) is a factor of that polynomial. Using this theorem, we can construct the factors for the given zeros. The factors corresponding to the zeros 4, √7, and -√7 are (x - 4), (x - √7), and (x + √7), respectively.

Now we need to consider the given options. To find the polynomial that has the specified zeros, we should look for the polynomial that can be factored into (x - 4)(x - √7)(x + √7). Note that (x - √7)(x + √7) is a difference of squares, which simplifies to x² - 7. So the expected polynomial is x³ - 4x² + 7x - 28, which factorizes to (x - 4)(x² - 7).

Comparing the given options, we can conclude that the correct polynomial function is 1) f(x) = x³ - 4x² + 7x - 28.

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