Question

find the zeros of the polynomial function.

\\(f(x) = 4(x - 6)(x + 7)^3\\)

a. \\(x = 6, x = -7\\)
b. \\(x = -6, x = 3\\)
c. \\(x = -6, x = 7, x = 3\\)
d. \\(x = 6, x = 3\\)

Explanation:

Set the polynomial function to zero

To find the zeros of a function, we set \(f(x) = 0\).
Using the Polynomial Functions concept:

$$4(x - 6)(x + 7)^3 = 0$$

Apply the zero product property

Set each distinct variable factor equal to zero.

$$x - 6 = 0 \quad \text{or} \quad (x + 7)^3 = 0$$

Solve each linear equation

Solve for \(x\) in both cases.

$$x = 6$$

$$x + 7 = 0 \implies x = -7$$

Identify the zeros

The zeros of the function are \(x = 6\) and \(x = -7\).
This matches option A.

Answer:

• (A) \(x = 6, x = -7\) (Correct answer)
• (B) \(x = -6, x = 3\)
• (C) \(x = -6, x = 7, x = 3\)
• (D) \(x = 6, x = 3\)