Question

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

Answer 1

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

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

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

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