Select the correct answer.

Which statement best describes the zeros of the function [tex]$g(x) = \left(x^2 - 1\right)\left(x^2 - 2x + 1\right)$[/tex]?

A. The function has two distinct real zeros.
B. The function has three distinct real zeros.
C. The function has four distinct real zeros.
D. The function has two distinct real zeros and two complex zeros.



Answer :

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.