Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

How many roots are there for the polynomial that factors into [tex]$f(x)=\left(x^2-1\right)(x+2)(x-1)^2$[/tex]?

A. 4
B. 3
C. 5
D. 6

Sagot :

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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.