Question:
If \[ x=-2 \] find the value of:
\[ \left(\frac{2}{x} – \frac{x}{2}\right) \left(\frac{4}{x^2} + \frac{x^2}{4} + 1\right) \]
Solution:
Rearranging the terms:
\[ \left(\frac{2}{x} – \frac{x}{2}\right) \left(\frac{4}{x^2} + \frac{2}{x}\cdot\frac{x}{2} + \frac{x^2}{4}\right) \]
Using identity:
\[ (a-b)(a^2+ab+b^2)=a^3-b^3 \]
Here, \[ a=\frac{2}{x},\qquad b=\frac{x}{2} \]
\[ = \left(\frac{2}{x}\right)^3 – \left(\frac{x}{2}\right)^3 \]
Substituting \[ x=-2 \]
\[ = \left(\frac{2}{-2}\right)^3 – \left(\frac{-2}{2}\right)^3 \]
\[ = (-1)^3-(-1)^3 \]
\[ =-1+1 \]
\[ =0 \]