To solve the given equation
[tex]\[
x^4 - \frac{11}{5} x^3 - 1 = \frac{1}{x} - 3
\][/tex]
we start by eliminating the fraction on the right-hand side. Multiply both sides by [tex]\(x\)[/tex]:
[tex]\[
x^5 - \frac{11}{5} x^4 - x = 1 - 3x
\][/tex]
Next, move all terms to one side to set the equation to 0:
[tex]\[
x^5 - \frac{11}{5} x^4 - x - 1 + 3x = 0
\][/tex]
Combine like terms:
[tex]\[
x^5 - \frac{11}{5} x^4 + 2x - 1 = 0
\][/tex]
This is a polynomial equation in [tex]\(x\)[/tex]. To solve this polynomial equation, we use methods such as factoring, the Rational Root Theorem, or numerical methods if it cannot be easily factored.
By solving this polynomial equation, the solutions are:
1. [tex]\( x \approx 0.610214422393071 \)[/tex]
2. [tex]\( x \approx 0.843913689056067 \)[/tex]
3. [tex]\( x \approx 2.01658735840272 \)[/tex]
4. [tex]\( x \approx -0.63535773492593 - 0.747842039672421i \)[/tex] (a complex solution)
5. [tex]\( x \approx -0.63535773492593 + 0.747842039672421i \)[/tex] (a complex solution)
The numerical and complex solutions indicate the roots of the polynomial equation that satisfy the original equation.
