\(\frac{x^5+x+1}{x^3-1}=\frac{x^5-x^2+x^2+x+1}{x^3-1}=\frac{x^2\left(x^3-1\right)+\left(x^2+x+1\right)}{x^3-1}\)

\(=\frac{x^2\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{\left(x^2+x+1\right)\left[x^2\left(x-1\right)+1\right]}{\left(x-1\right)\left(x^2+x+1\right)}\)

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

a,\(A=\frac{6x+12}{\left(x+2\right)\left(2x-6\right)}=\frac{6\left(x+2\right)}{2\left(x+2\right)\left(x-3\right)}=\frac{3}{x-3}\)

b, Giá trị của x để phân thức có giá trị bằng (-2) : 

\(\frac{3}{x-3}=-2\Rightarrow x=1,5\)

Ai giúp mình câu 2 với

Cau 2:

A= \(\frac{x}{x+1}.\frac{x+1}{x+2}.\frac{x+2}{x+3}.\frac{x+3}{x+4}.\frac{x+4}{x+5}.\frac{x+5}{x+6}.\frac{x+6}{x+7}\)

A= \(\frac{x}{x+7}\)

\(\frac{x^4+x^3+x+1}{x^4-x^3+2x^2-x+1}\)

\(=\frac{x\left(x+1\right)+\left(x+1\right)}{x\left(x-1\right)+2x^2-2x+x+1}\)

\(=\frac{\left(x+1\right)\left(x+1\right)}{x\left(x-1\right)+2\left(x-1\right)+\left(x+1\right)}\)

Ddeeff sao rồi bạn ko rút gọn được

Cả tử và mẫu có nhân tử chung là x² + x + 1 rút gọn cái đó đi là được

\(\frac{1}{x\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}+...+\frac{1}{\left(x-4\right)\left(x-5\right)}\)

\(=\frac{1}{x}-\frac{1}{x-1}+\frac{1}{x-1}-\frac{1}{x-2}+\frac{1}{x-2}-\frac{1}{x-3}+...+\frac{1}{x-4}-\frac{1}{x-5}\)

\(=\frac{1}{x}-\frac{1}{x-5}=\frac{x-5}{x\left(x-5\right)}-\frac{x}{x\left(x-5\right)}=\frac{-5}{x\left(x-5\right)}\)

\(\frac{1}{x\left(x-1\right)}+\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}+...+\frac{1}{\left(x-4\right)\left(x-5\right)}\)

\(=\frac{1}{x}-\frac{1}{x-1}+\frac{1}{x-1}-\frac{1}{x-2}+...+\frac{1}{x-4}-\frac{1}{x-5}\)

\(=\frac{1}{x}-\frac{1}{x-5}\)

\(=\frac{x-5}{x\left(x-5\right)}-\frac{x}{x\left(x-5\right)}\)

\(=\frac{x-5-x}{x\left(x-5\right)}\)

\(=-\frac{5}{x\left(x-5\right)}\)

Bài 10 trang 40 sgk toán lớp 8 tập ko 

\(\frac{x^7+x^6+x^5+x^4+x^3+x^2+x+1}{x^2-1}\left(DK:x\ne-1;x\ne1\right)\)

\(=\frac{x^4\left(x^3+x^2+x+1\right)+\left(x^3+x^2+x+1\right)}{x^2-1}\)

\(=\frac{x^4\left[x\left(x^2+1\right)+x^2+1\right]+\left[x\left(x^2+1\right)+x^2+1\right]}{x^2-1}\)

\(=\frac{\left(x^4+1\right)\left(x+1\right)\left(x^2+1\right)}{\left(x-1\right)\left(x+1\right)}=\frac{\left(x^2+1\right)\left(x^4+1\right)}{x-1}\)

\(\frac{x^7+x^6+x^5+x^4+x^3+x^2+x+1}{x^2-1}\)

\(=\frac{x^6\left(x+1\right)+x^4\left(x+1\right)+x^2\left(x+1\right)+\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)

\(=\frac{\left(x+1\right)\left(x^6+x^4+x^2\right)}{\left(x+1\right)\left(x-1\right)}\)

\(=\frac{x^6+x^4+x^2}{x+1}\)

\(=\frac{x^2\left(x^3+x^2+1\right)}{x+1}\)