To solve the given equation [tex]\(x^5 - x = 10\)[/tex], we need to find the roots (values of [tex]\(x\)[/tex]) that satisfy the equation. The equation can be rewritten in the standard form:
[tex]\[ x^5 - x - 10 = 0 \][/tex]
Now, let's find the roots of this polynomial equation. The roots of the equation [tex]\( x^5 - x - 10 = 0 \)[/tex] are the values of [tex]\( x \)[/tex] that make the left-hand side zero. After solving the equation, we obtain five distinct roots:
1. [tex]\( x_0 \)[/tex] such that [tex]\( x_0^5 - x_0 - 10 = 0 \)[/tex]
2. [tex]\( x_1 \)[/tex] such that [tex]\( x_1^5 - x_1 - 10 = 0 \)[/tex]
3. [tex]\( x_2 \)[/tex] such that [tex]\( x_2^5 - x_2 - 10 = 0 \)[/tex]
4. [tex]\( x_3 \)[/tex] such that [tex]\( x_3^5 - x_3 - 10 = 0 \)[/tex]
5. [tex]\( x_4 \)[/tex] such that [tex]\( x_4^5 - x_4 - 10 = 0 \)[/tex]
In mathematical notation using the concept of complex roots [tex]\( CRootOf \)[/tex], these roots can be written as:
[tex]\[ x_0 = CRootOf(x^5 - x - 10, 0) \][/tex]
[tex]\[ x_1 = CRootOf(x^5 - x - 10, 1) \][/tex]
[tex]\[ x_2 = CRootOf(x^5 - x - 10, 2) \][/tex]
[tex]\[ x_3 = CRootOf(x^5 - x - 10, 3) \][/tex]
[tex]\[ x_4 = CRootOf(x^5 - x - 10, 4) \][/tex]
Each [tex]\( CRootOf \)[/tex] represents a unique root of the equation. The [tex]\(CRootOf\)[/tex] function indicates that these roots are solutions to the polynomial equation [tex]\(x^5 - x - 10 = 0\)[/tex].
These roots are typically complex numbers (they may have both real and imaginary parts) and can be found using numerical methods or specialized algebraic software. The roots are distinct, which reflects the nature of a fifth-degree polynomial having up to five unique solutions.
Thus, the complete set of roots for the equation [tex]\( x^5 - x - 10 = 0 \)[/tex] is:
[tex]\[ [CRootOf(x^5 - x - 10, 0), CRootOf(x^5 - x - 10, 1), CRootOf(x^5 - x - 10, 2), CRootOf(x^5 - x - 10, 3), CRootOf(x^5 - x - 10, 4)] \][/tex]
These are the solutions for the given polynomial equation.
