Упростите вырожение [tex](\frac{1}{x-1}-\frac{x+1}{x^{2}+x+1}):(1+\frac{1}{x^{3}-1}[/tex]). Created by Юля11. algebra-ru

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triolana

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Svet1ana

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$(\frac{1}{x-1}-\frac{x+1}{x^{2}+x+1}):(1+\frac{1}{x^{3}-1})=\frac{x+2}{x^{3}}$

1) $\frac{1}{x-1}-\frac{x+1}{x^{2}+x+1}=\frac{x^{2}+x+1-(x+1)(x-1)}{(x-1)(x^{2}+x+1)}= \frac{x^{2}+x+1-(x^{2}-x+x-1)}{(x-1)(x^{2}+x+1)}=\frac{x^{2}+x+1-(x^{2}+(-x+x)-1)}{(x-1)(x^{2}+x+1)}=$ $\frac{x^{2}+x+1-(x^{2}-1)}{x^{3}-1}=\frac{x^{2}+x+1-x^{2}+1}{x^{3}-1}=\frac{(x^{2}-x^{2})+x+(1+1)}{x^{3}-1}=\frac{x+2}{x^{3}-1}$

2) $1+\frac{1}{x^{3}-1}=\frac{x^{3}-1+1}{x^{3}-1}=\frac{x^{3}+(-1+1)}{x^{3}-1}=\frac{x^{3}}{x^{3}-1}$

3) $\frac{x+2}{x^{3}-1}:\frac{x^{3}}{x^{3}-1}=\frac{x+2}{x^{3}-1}\cdot\frac{x^{3}-1}{x^{3}}=\frac{x+2}{x^{3}}$

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