Let's analyze the function [tex]\( g(x) = (x^2 - 1)(x^2 - 2x + 1) \)[/tex] to determine its zeros.
1. Simplify [tex]\( g(x) \)[/tex]:
First, observe that the function is a product of two simpler quadratic functions:
[tex]\[
g(x) = (x^2 - 1)(x^2 - 2x + 1)
\][/tex]
2. Find the zeros of each factor:
- The first factor is [tex]\( x^2 - 1 \)[/tex]. This is a difference of squares and can be factored further:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
The zeros are:
[tex]\[
x = 1 \quad \text{and} \quad x = -1
\][/tex]
- The second factor is [tex]\( x^2 - 2x + 1 \)[/tex]. This is a perfect square trinomial and can be factored as:
[tex]\[
x^2 - 2x + 1 = (x - 1)^2
\][/tex]
The zero is:
[tex]\[
x = 1
\][/tex]
Note that [tex]\( x = 1 \)[/tex] is a double root.
3. Combine the zeros:
From the factors, we have identified the zeros:
[tex]\[
x = -1, \quad x = 1 \, (\text{with multiplicity 3})
\][/tex]
4. Determine the number of distinct zeros:
The distinct zeros are:
[tex]\[
x = -1 \quad \text{and} \quad x = 1
\][/tex]
Even though [tex]\( x = 1 \)[/tex] appears multiple times in the factors, it is counted as a single distinct zero.
5. Count the distinct zeros:
We have:
[tex]\[
x = -1 \quad \text{(1 distinct zero)}
\][/tex]
[tex]\[
x = 1 \quad \text{(1 distinct zero)}
\][/tex]
Therefore, there are two distinct real zeros.
The correct answer is:
- The function has two distinct real zeros.
