Question

Упростите вырожение $(frac{1}{x-1}-frac{x+1}{x^{2}+x+1}):(1+frac{1}{x^{3}-1}$)

Answer 1

решние во вложении

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b2f3e34b10101f3fdf827c4f955179bf.doc

Answer 2

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

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

b2f3e34b10101f3fdf827c4f955179bf.doc