Question

C=\(\dfrac{8-12x-6x^2-x^2}{x^3-2x^2+x-2}\)

CMR C<0

Answer 1

Mình nghĩ đề bài là:

CMR : \(C=\frac{8-12x+6x^2-x^3}{x^3-2x^2+x-2}< 0\)

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ĐK: \(x^3-2x^2+x-2\neq 0\)

\(\Leftrightarrow x^2(x-2)+(x-2)\neq 0\)

\(\Leftrightarrow (x-2)(x^2+1)\neq 0\Rightarrow x\neq 2\)

Ta có: \(C=\frac{8-12x+6x^2-x^3}{x^3-2x^2+x-2}=-\frac{x^3-6x^2+12x-8}{(x^2+1)(x-2)}\)

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

Với mọi \(x\neq 2\Rightarrow (x-2)^2>0\), mà \(x^2+1>0, \forall x\in\mathbb{R}\)

\(\Rightarrow \frac{(x-2)^2}{x^2+1}>0\Rightarrow C=-\frac{(x-2)^2}{x^2+1}< 0\) (đpcm)