Given [tex]\( x = \left(\frac{5 - \sqrt{2}}{2}\right) \)[/tex], let's break down the problem step by step to find the value of the expression:
[tex]\[ \left(x^3 + \frac{1}{x^3}\right) - 5\left(x^3 + \frac{1}{x^2}\right) + \left(x + \frac{1}{x}\right) \][/tex]
1. Calculate [tex]\( x^3 \)[/tex]:
[tex]\[
x^3 \approx 5.7632
\][/tex]
2. Calculate [tex]\( \frac{1}{x^3} \)[/tex]:
[tex]\[
\frac{1}{x^3} \approx 0.1735
\][/tex]
3. Calculate [tex]\( x^3 + \frac{1}{x^3} \)[/tex]:
[tex]\[
x^3 + \frac{1}{x^3} \approx 5.7632 + 0.1735 = 5.9367
\][/tex]
4. Calculate [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[
\frac{1}{x^2} \approx 0.3111
\][/tex]
5. Calculate [tex]\( x^3 + \frac{1}{x^2} \)[/tex]:
[tex]\[
x^3 + \frac{1}{x^2} \approx 5.7632 + 0.3111 = 6.0743
\][/tex]
6. Multiply this result by 5:
[tex]\[
5 \left( x^3 + \frac{1}{x^2} \right) \approx 5 \times 6.0743 = 30.3714
\][/tex]
7. Calculate [tex]\( \frac{1}{x} \)[/tex]:
[tex]\[
\frac{1}{x} \approx 0.5578
\][/tex]
8. Calculate [tex]\( x + \frac{1}{x} \)[/tex]:
[tex]\[
x + \frac{1}{x} \approx 1.7929 + 0.5578 = 2.3507
\][/tex]
Finally, combine all the parts of the expression:
[tex]\[
(x^3 + \frac{1}{x^3}) - 5(x^3 + \frac{1}{x^2}) + (x + \frac{1}{x}) \approx 5.9367 - 30.3714 + 2.3507 = -22.0841
\][/tex]
Thus, the value of the expression is approximately
[tex]\[
-22.0841
\][/tex]
