How do you find all the zeros of the function and write the polynomial as a product of linear factors for f(x) = x² – 25?

To find the zeros of the function f(x) = x² – 25, we start by setting the function equal to zero:

f(x) = 0
x² – 25 = 0

Next, we can solve for x by adding 25 to both sides:

x² = 25

Now, we take the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution:

x = ±√25
x = ±5

This gives us the zeros of the function: x = 5 and x = -5.

Now, to express the polynomial as a product of linear factors, we can use the zeros we found:

f(x) = (x – 5)(x + 5)

This factorization shows the polynomial x² – 25 as a product of its linear factors. Therefore, the complete answer is:

Zeros: x = 5, x = -5
Polynomial in linear factors: (x – 5)(x + 5)