Answer :
Certainly! Let's solve the equation [tex]\( x^9 - 3x + 2 = 0 \)[/tex].
### Step-by-Step Solution:
1. Understanding the Equation:
The given equation is a polynomial equation of degree 9:
[tex]\[ x^9 - 3x + 2 = 0 \][/tex]
This means we are looking for the values of [tex]\( x \)[/tex] that satisfy this equation.
2. Identifying Solutions:
The nature of polynomials suggests that we can have up to 9 roots (solutions), which might be real or complex.
3. Finding Solutions:
Upon solving the polynomial equation, we obtain the following solutions:
[tex]\[ x = 1, \quad \text{ and } \quad \text{8 distinct roots of a 8-degree polynomial } \][/tex]
Specifically, these roots are:
[tex]\[ CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, k) \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6, \text{and} 7 \][/tex]
Here, [tex]\( CRootOf \)[/tex] denotes complex roots of another polynomial.
### Explanation of the Roots:
1. The Simple Root:
- [tex]\( x = 1 \)[/tex] is a straightforward solution, meaning [tex]\( x = 1 \)[/tex] satisfies the equation directly.
2. Complex Roots:
- The more complex roots are given by solving the auxiliary polynomial:
[tex]\[ x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2 = 0 \][/tex]
- Each [tex]\( k \)[/tex] in [tex]\( CRootOf \)[/tex] represents a distinct root of this polynomial.
### Conclusion:
The complete set of solutions for the equation [tex]\( x^9 - 3x + 2 = 0 \)[/tex] is:
[tex]\[ x = 1, \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 0), \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 1), \quad \ldots \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 7) \][/tex]
This means [tex]\( x = 1 \)[/tex] and the other 8 roots are derived from the auxiliary polynomial [tex]\( x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2 \)[/tex].
### Step-by-Step Solution:
1. Understanding the Equation:
The given equation is a polynomial equation of degree 9:
[tex]\[ x^9 - 3x + 2 = 0 \][/tex]
This means we are looking for the values of [tex]\( x \)[/tex] that satisfy this equation.
2. Identifying Solutions:
The nature of polynomials suggests that we can have up to 9 roots (solutions), which might be real or complex.
3. Finding Solutions:
Upon solving the polynomial equation, we obtain the following solutions:
[tex]\[ x = 1, \quad \text{ and } \quad \text{8 distinct roots of a 8-degree polynomial } \][/tex]
Specifically, these roots are:
[tex]\[ CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, k) \quad \text{for} \quad k = 0, 1, 2, 3, 4, 5, 6, \text{and} 7 \][/tex]
Here, [tex]\( CRootOf \)[/tex] denotes complex roots of another polynomial.
### Explanation of the Roots:
1. The Simple Root:
- [tex]\( x = 1 \)[/tex] is a straightforward solution, meaning [tex]\( x = 1 \)[/tex] satisfies the equation directly.
2. Complex Roots:
- The more complex roots are given by solving the auxiliary polynomial:
[tex]\[ x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2 = 0 \][/tex]
- Each [tex]\( k \)[/tex] in [tex]\( CRootOf \)[/tex] represents a distinct root of this polynomial.
### Conclusion:
The complete set of solutions for the equation [tex]\( x^9 - 3x + 2 = 0 \)[/tex] is:
[tex]\[ x = 1, \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 0), \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 1), \quad \ldots \quad CRootOf(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2, 7) \][/tex]
This means [tex]\( x = 1 \)[/tex] and the other 8 roots are derived from the auxiliary polynomial [tex]\( x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x - 2 \)[/tex].
