Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine the zeros of the function [tex]\( g(x) = (x^2 - 1)(x^2 - 2x + 1) \)[/tex], let's proceed step by step by examining each factor separately.
1. First factor: [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ x^2 - 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x^2 = 1 \][/tex]
Taking the square root of both sides:
[tex]\[ x = \pm 1 \][/tex]
So, the zeros of the first factor are [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
2. Second factor: [tex]\( x^2 - 2x + 1 \)[/tex]:
[tex]\[ x^2 - 2x + 1 = 0 \][/tex]
Notice that this can be factored as a perfect square:
[tex]\[ (x - 1)^2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 1 = 0 \implies x = 1 \][/tex]
So, the zero of the second factor is [tex]\( x = 1 \)[/tex].
3. Combining the zeros:
From the first factor, we have the zeros [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex]. From the second factor, we have the zero [tex]\( x = 1 \)[/tex].
Combining these zeros, the complete set of zeros is:
[tex]\[ \{-1, 1, 1\} \][/tex]
4. Identifying distinct zeros:
The distinct zeros among [tex]\(-1, 1, 1\)[/tex] are simply [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
5. Counting the distinct zeros:
There are 2 distinct real zeros: [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
Hence, the correct statement is:
- The function has two distinct real zeros.
1. First factor: [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ x^2 - 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x^2 = 1 \][/tex]
Taking the square root of both sides:
[tex]\[ x = \pm 1 \][/tex]
So, the zeros of the first factor are [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].
2. Second factor: [tex]\( x^2 - 2x + 1 \)[/tex]:
[tex]\[ x^2 - 2x + 1 = 0 \][/tex]
Notice that this can be factored as a perfect square:
[tex]\[ (x - 1)^2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 1 = 0 \implies x = 1 \][/tex]
So, the zero of the second factor is [tex]\( x = 1 \)[/tex].
3. Combining the zeros:
From the first factor, we have the zeros [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex]. From the second factor, we have the zero [tex]\( x = 1 \)[/tex].
Combining these zeros, the complete set of zeros is:
[tex]\[ \{-1, 1, 1\} \][/tex]
4. Identifying distinct zeros:
The distinct zeros among [tex]\(-1, 1, 1\)[/tex] are simply [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
5. Counting the distinct zeros:
There are 2 distinct real zeros: [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
Hence, the correct statement is:
- The function has two distinct real zeros.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.