Question

find all real zeros of the function.

\\f(x) = 5x(x - 1)^2(x - 7)^2\\

if there is more than one answer, separate them with commas.

zero(s):

Explanation:

Set the function to zero

To find the real zeros, set \(f(x) = 0\).

$$5x(x - 1)^2(x - 7)^2 = 0$$

Apply the zero product property

Set each distinct factor to zero.

$$5x = 0 \quad\text{or}\quad (x - 1)^2 = 0 \quad\text{or}\quad (x - 7)^2 = 0$$

Solve each individual equation

Solve for \(x\) in each case.

$$x = 0$$

$$x - 1 = 0 \implies x = 1$$

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

List the distinct real zeros

Combine the unique real solutions.

$$x = 0, 1, 7$$

Answer:

zero(s): <blank>0, 1, 7</blank>