Question

Cho phân thức A = \(\frac{x^4+x^3+x+1}{x^4-x^3+2x^2-x+1}\).

a . Rút gọn A .

b . Chứng minh A luôn không âm với mọi x .

Answer 1

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

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

Do \(\left\{{}\begin{matrix}\left(x+1\right)^2\ge0\\x^2+1>0\end{matrix}\right.\) \(\forall x\Rightarrow A\ge0\) \(\forall x\)