To solve the polynomial equation [tex]\(-5x^5 + 7x^4 - 20x^2 - 2x - 9 = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] that satisfy this equation. Given the complexity of this polynomial, we'll follow steps to solve it symbolically as it is not easily factorable by simple methods such as factoring by inspection or using the Rational Root Theorem. For higher-degree polynomials like this fifth-degree polynomial, numerical methods or computational algebra systems offer a more practical approach to finding roots.
### Step-by-Step Process:
1. Set Up the Equation:
Start with the polynomial equation:
[tex]\[ -5x^5 + 7x^4 - 20x^2 - 2x - 9 = 0 \][/tex]
2. Identify Possible Solutions:
Polynomial equations of degree five generally have up to five solutions, which can be real or complex. Here, we will denote the solutions as roots of the polynomial.
3. Use Symbolic Solution Techniques:
We employ symbolic algebra to solve the polynomial equation. This involves finding the roots of the polynomial, which can be represented in terms of algebraic objects that capture both real and complex roots.
4. Obtain the Roots:
Solving this polynomial equation, we obtain the roots, referred to as CRootOf objects (which are algebraic numbers). These are complex roots due to the nature of polynomial equations of degree greater than four. The solutions are represented as:
[tex]\[
\begin{align*}
&\text{Root 1:} \quad \text{CRootOf}(5x^5 - 7x^4 + 20x^2 + 2x + 9, 0) \\
&\text{Root 2:} \quad \text{CRootOf}(5x^5 - 7x^4 + 20x^2 + 2x + 9, 1) \\
&\text{Root 3:} \quad \text{CRootOf}(5x^5 - 7x^4 + 20x^2 + 2x + 9, 2) \\
&\text{Root 4:} \quad \text{CRootOf}(5x^5 - 7x^4 + 20x^2 + 2x + 9, 3) \\
&\text{Root 5:} \quad \text{CRootOf}(5x^5 - 7x^4 + 20x^2 + 2x + 9, 4)
\end{align*}
\][/tex]
### Conclusion:
The polynomial [tex]\(-5x^5 + 7x^4 - 20x^2 - 2x - 9 = 0\)[/tex] has five roots, which are represented as complex algebraic numbers known as CRootOf objects. These roots encapsulate the solutions to the polynomial and can be numerically approximated if necessary.
Understanding these roots symbolically allows us to work with the polynomial's solutions, providing a rigorous and complete description of all solutions.
