At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Sure! Let's find the roots of the polynomial [tex]\( f(x) = x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex] given that some of the roots include 1 and [tex]\( i \)[/tex].
### Step-by-Step Solution:
1. Given roots:
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = i \)[/tex]
Since polynomial coefficients are real numbers, the complex conjugate of any non-real roots must also be a root. Therefore, [tex]\( x = -i \)[/tex] is also a root.
2. Polynomial and its Roots:
- We know the given polynomial has [tex]\( x = 1 \)[/tex], [tex]\( x = i \)[/tex], and [tex]\( x = -i \)[/tex] as roots.
3. Construct the corresponding factors:
- For [tex]\( x = 1 \)[/tex]: [tex]\( (x - 1) \)[/tex]
- For [tex]\( x = i \)[/tex] and [tex]\( x = -i \)[/tex]: The corresponding factor is [tex]\( (x - i)(x + i) = x^2 + 1 \)[/tex]
4. Form the partially factored polynomial:
- Combine the factors we have: [tex]\( (x - 1)(x^2 + 1) \)[/tex]
5. Perform polynomial division:
To find the remaining polynomial after factoring out [tex]\( (x - 1)(x^2 + 1) \)[/tex] from [tex]\( x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex], we need to perform polynomial division on [tex]\( f(x) = x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex]:
- First, divide [tex]\( f(x) \)[/tex] by [tex]\( (x - 1) \)[/tex]:
[tex]\[ f(x) \div (x - 1) \][/tex]
This will give us a quotient polynomial.
Let's denote this quotient by [tex]\( Q_1(x) \)[/tex]:
[tex]\[ Q_1(x) = x^4 - 2x^3 + 6x^2 - 2x + 5 \][/tex]
- Next, divide the obtained quotient [tex]\( Q_1(x) \)[/tex] by [tex]\( (x^2 + 1) \)[/tex]:
[tex]\[ Q_1(x) \div (x^2 + 1) \][/tex]
This will give us another quotient polynomial.
Let's denote the new quotient by [tex]\( Q_2(x) \)[/tex]:
[tex]\[ Q_2(x) = x^2 - 2x + 5 \][/tex]
6. Solving the remaining polynomial [tex]\( Q_2(x) = x^2 - 2x + 5 \)[/tex]:
[tex]\[ x^2 - 2x + 5 = 0 \][/tex]
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm \sqrt{-16}}{2} = \frac{2 \pm 4i}{2} = 1 \pm 2i \][/tex]
The remaining roots are [tex]\( x = 1 + 2i \)[/tex] and [tex]\( x = 1 - 2i \)[/tex].
### In summary:
The roots of the polynomial [tex]\( x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex] are:
1. [tex]\( x = 1 \)[/tex]
2. [tex]\( x = i \)[/tex]
3. [tex]\( x = -i \)[/tex]
4. [tex]\( x = 1 + 2i \)[/tex]
5. [tex]\( x = 1 - 2i \)[/tex]
### Step-by-Step Solution:
1. Given roots:
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = i \)[/tex]
Since polynomial coefficients are real numbers, the complex conjugate of any non-real roots must also be a root. Therefore, [tex]\( x = -i \)[/tex] is also a root.
2. Polynomial and its Roots:
- We know the given polynomial has [tex]\( x = 1 \)[/tex], [tex]\( x = i \)[/tex], and [tex]\( x = -i \)[/tex] as roots.
3. Construct the corresponding factors:
- For [tex]\( x = 1 \)[/tex]: [tex]\( (x - 1) \)[/tex]
- For [tex]\( x = i \)[/tex] and [tex]\( x = -i \)[/tex]: The corresponding factor is [tex]\( (x - i)(x + i) = x^2 + 1 \)[/tex]
4. Form the partially factored polynomial:
- Combine the factors we have: [tex]\( (x - 1)(x^2 + 1) \)[/tex]
5. Perform polynomial division:
To find the remaining polynomial after factoring out [tex]\( (x - 1)(x^2 + 1) \)[/tex] from [tex]\( x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex], we need to perform polynomial division on [tex]\( f(x) = x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex]:
- First, divide [tex]\( f(x) \)[/tex] by [tex]\( (x - 1) \)[/tex]:
[tex]\[ f(x) \div (x - 1) \][/tex]
This will give us a quotient polynomial.
Let's denote this quotient by [tex]\( Q_1(x) \)[/tex]:
[tex]\[ Q_1(x) = x^4 - 2x^3 + 6x^2 - 2x + 5 \][/tex]
- Next, divide the obtained quotient [tex]\( Q_1(x) \)[/tex] by [tex]\( (x^2 + 1) \)[/tex]:
[tex]\[ Q_1(x) \div (x^2 + 1) \][/tex]
This will give us another quotient polynomial.
Let's denote the new quotient by [tex]\( Q_2(x) \)[/tex]:
[tex]\[ Q_2(x) = x^2 - 2x + 5 \][/tex]
6. Solving the remaining polynomial [tex]\( Q_2(x) = x^2 - 2x + 5 \)[/tex]:
[tex]\[ x^2 - 2x + 5 = 0 \][/tex]
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm \sqrt{-16}}{2} = \frac{2 \pm 4i}{2} = 1 \pm 2i \][/tex]
The remaining roots are [tex]\( x = 1 + 2i \)[/tex] and [tex]\( x = 1 - 2i \)[/tex].
### In summary:
The roots of the polynomial [tex]\( x^5 - 3x^4 + 8x^3 - 8x^2 + 7x - 5 \)[/tex] are:
1. [tex]\( x = 1 \)[/tex]
2. [tex]\( x = i \)[/tex]
3. [tex]\( x = -i \)[/tex]
4. [tex]\( x = 1 + 2i \)[/tex]
5. [tex]\( x = 1 - 2i \)[/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
