To determine how many roots the polynomial [tex]\( f(x) = \left(x^2 - 1\right)(x + 2)(x - 1)^2 \)[/tex] has, we need to analyze the factors of the polynomial and find the roots.
1. Factor Analysis:
- [tex]\( x^2 - 1 \)[/tex] can be factored as [tex]\( (x - 1)(x + 1) \)[/tex].
- [tex]\( x + 2 \)[/tex] remains unchanged.
- [tex]\( (x - 1)^2 \)[/tex].
So, the polynomial can be written as:
[tex]\[ f(x) = (x - 1)(x + 1)(x + 2)(x - 1)^2 \][/tex]
2. Finding the Roots:
- Set each factor equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ x - 1 = 0 \Rightarrow x = 1 \][/tex]
[tex]\( x = 1 \)[/tex] appears from both [tex]\( (x - 1) \)[/tex] and [tex]\( (x - 1)^2 \)[/tex], so it has a multiplicity.
[tex]\[ x + 1 = 0 \Rightarrow x = -1 \][/tex]
[tex]\[ x + 2 = 0 \Rightarrow x = -2 \][/tex]
[tex]\( (x - 1)^2 = 0 \Rightarrow x = 1 \)[/tex]. This confirms [tex]\( x = 1 \)[/tex] with a multiplicity of 2.
3. Counting the Roots:
- The roots are: [tex]\( 1 \)[/tex] (from [tex]\( (x - 1) \)[/tex] and [tex]\( (x - 1)^2 \)[/tex]), [tex]\( -1 \)[/tex] (from [tex]\( x + 1 \)[/tex]), and [tex]\( -2 \)[/tex] (from [tex]\( x + 2 \)[/tex]).
- The multiplicity of [tex]\( 1 \)[/tex] is 2 due to [tex]\( (x - 1)(x - 1)^2 \)[/tex].
The roots can be listed with their multiplicities:
[tex]\[ x = 1, -1, -2, 1 \][/tex] (counting [tex]\( x = 1 \)[/tex] twice due to multiplicity).
Therefore, the total number of roots, counting multiplicities, is 5.
The correct answer is:
[tex]\[
5
\][/tex]
