To solve the given expression involving the value [tex]\( x = \left(\frac{5 - \sqrt{21}}{2}\right) \)[/tex], we will need to compute various parts and then combine them according to the given formula. The expression we need to evaluate is:
[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]
First, we will determine the value of [tex]\( x \)[/tex].
[tex]\[ x = \left(\frac{5 - \sqrt{21}}{2}\right) \][/tex]
Next, we calculate [tex]\( x^3 \)[/tex]:
[tex]\[ x^3 \approx 0.009091660529919932 \][/tex]
We then find the reciprocal of [tex]\( x \)[/tex]:
[tex]\[ \frac{1}{x} \approx 4.791287847477918 \][/tex]
Now, we calculate the cube of the reciprocal:
[tex]\[ \left(\frac{1}{x}\right)^3 \approx 109.99090833946993 \][/tex]
We then sum [tex]\( x^3 \)[/tex] and [tex]\( \left(\frac{1}{x}\right)^3 \)[/tex]:
[tex]\[ x^3 + \frac{1}{x^3} \approx 0.009091660529919932 + 109.99090833946993 = 109.99999999999984 \][/tex]
Then, calculate [tex]\( x + \frac{1}{x} \)[/tex]:
[tex]\[ x + \frac{1}{x} \approx 0.20871215252208009 + 4.791287847477918 = 4.999999999999998 \][/tex]
Combining these parts with the original formula, we have:
[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]
[tex]\[ \approx 109.99999999999984 - 5 \left(109.99999999999984\right) + 4.999999999999998 \][/tex]
Calculate the intermediate terms:
[tex]\[ 5 \left(x^3 + \frac{1}{x^3}\right) \approx 5 \times 109.99999999999984 = 549.9999999999992 \][/tex]
Finally, evaluate the overall expression:
[tex]\[ 109.99999999999984 - 549.9999999999992 + 4.999999999999998 \][/tex]
[tex]\[ = -434.9999999999994 \][/tex]
Thus, 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] is:
[tex]\[ \boxed{-434.9999999999994} \][/tex]
