Given that [tex]\( x = \left( \frac{5 - \sqrt{2}}{2} \right) \)[/tex], we need to find the value of the expression:
[tex]\[
\left(x^3 + \frac{1}{x^3}\right) - 5\left(x^2 + \frac{1}{x^2}\right) + \left(x + \frac{1}{x}\right)
\][/tex]
Let's break it down step-by-step:
### Step 1: Calculate [tex]\( x^3 \)[/tex] and [tex]\( \frac{1}{x^3} \)[/tex]
First, evaluate [tex]\( x^3 \)[/tex]:
[tex]\[
x^3 = 5.76319446215896
\][/tex]
Then, evaluate [tex]\( \frac{1}{x^3} \)[/tex]:
[tex]\[
\frac{1}{x^3} = 0.17351488077766306
\][/tex]
### Step 2: Calculate [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex]
Next, evaluate [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = 3.2144660940672627
\][/tex]
Then, evaluate [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[
\frac{1}{x^2} = 0.31109365310949677
\][/tex]
### Step 3: Calculate [tex]\( x \)[/tex] and [tex]\( \frac{1}{x} \)[/tex]
Now, evaluate [tex]\( x \)[/tex]:
[tex]\[
x = 1.7928932188134525
\][/tex]
Then, evaluate [tex]\( \frac{1}{x} \)[/tex]:
[tex]\[
\frac{1}{x} = 0.5577577010759213
\][/tex]
### Step 4: Combine the results
Now, plug these values into the original expression:
[tex]\[
\left(x^3 + \frac{1}{x^3}\right) - 5\left(x^2 + \frac{1}{x^2}\right) + \left(x + \frac{1}{x}\right)
\][/tex]
Start with the cubic terms:
[tex]\[
x^3 + \frac{1}{x^3} = 5.76319446215896 + 0.17351488077766306 = 5.936709342936623
\][/tex]
Next, the quadratic terms multiplied by 5:
[tex]\[
5 \left( x^2 + \frac{1}{x^2} \right) = 5 \left( 3.2144660940672627 + 0.31109365310949677 \right) = 5 \times 3.5255597471767595 = 17.6277987358837975
\][/tex]
Then, the linear terms:
[tex]\[
x + \frac{1}{x} = 1.7928932188134525 + 0.5577577010759213 = 2.3506509198893738
\][/tex]
Finally, combine all the terms to find the result:
[tex]\[
\left( 5.936709342936623 \right) - \left( 17.6277987358837975 \right) + \left( 2.3506509198893738 \right) = -9.3404384730578
\][/tex]
Thus, the value of the given expression is:
[tex]\[
\boxed{-9.3404384730578}
\][/tex]
