1.4 Lösning 8c

Vi förenklar först uttryckets första term\[ {2\,x^2 - x^3 \over 2\,x^2 - 8} \; = \; {x^2\,(2 - x) \over 2\,(x^2 - 4)} \; = \; {x^2\,(2 - x) \over 2\,(x-2)\cdot(x+2)} \; = \; \]

\( = \; {-\,x^2\,(x - 2) \over 2\,(x-2)\cdot(x+2)} \; = \; {-\,x^2 \over 2\,(x+2)} \) Detta sätts in i hela uttrycket\[ {2\,x^2 - x^3 \over 2\,x^2 - 8} - {x \over x+2} + {x+2 \over 2} \; = \; {-\,x^2 \over 2\,(x+2)} - {x \over x+2} + {x+2 \over 2} \; = \; \]

\( = \; {-\,x^2 \over 2\,(x+2)} - {2\cdot x \over 2\cdot(x+2)} + {(x+2)\cdot(x+2) \over 2\cdot(x+2)} \; = \; \)

\( = \; {-\,x^2 - 2\,x + (x+2)^2 \over 2\,(x+2)} \; = \; {-\,x^2 - 2\,x + x^2 + 4\,x +4 \over 2\,(x+2)} \; = \; \)

\( = \; {2\,x + 4 \over 2\,(x+2)} \; = \; {2\,(x+2) \over 2\,(x+2)} \; = \; 1 \)