High School

Using the Factor Theorem, which of the polynomial functions has the zeros 4, \(\sqrt{7}\), and \(-\sqrt{7}\)?

A. \( f(x) = x^3 – 4x^2 + 7x + 28 \)
B. \( f(x) = x^3 – 4x^2 – 7x + 28 \)
C. \( f(x) = x^3 + 4x^2 – 7x + 28 \)
D. \( f(x) = x^3 + 4x^2 – 7x – 28 \)

Answer :

Final Answer:

By Using the Factor Theorem, f (x) = x³ – 4x² – 7x + 28 polynomial functions has the zeros 4, √7 , and -√7.

The correct option is b. f (x) = x³ – 4x² – 7x + 28.

Explanation:

The Factor Theorem states that if a polynomial function has a zero, then the corresponding factor is also present in the polynomial. In this case, the zeros are 4, √7, and -√7.

Let's break down the options and see which one matches the given zeros:

a. f (x) = x³ – 4x² + 7x + 28

b. f (x) = x³ – 4x² – 7x + 28

c. f (x) = x³ + 4x² – 7x + 28

d. f (x) = x³ + 4x² – 7x – 28

For a zero of 4, the factor is (x - 4).

For zeros of √7 and -√7, the factors are (x - √7) and (x + √7), respectively.

Now, let's multiply these factors:

(x - 4)(x - √7)(x + √7)

(x - 4)(x² - 7)

Expanding, we get:

x³ - 4x² - 7x + 28

This matches the polynomial function in option b. Thus, the polynomial function **f (x) = x³ – 4x² – 7x + 28** has the given zeros.

Learn more about Factor Theorem

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