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sua de \(\frac{3}{x^4-x^3+x-1}\) \(-\frac{1}{x^4+x^3-x-1}-\frac{4}{x^5-x^4+x^3-x^2+x-1}\) (dk \(x\ne+-1\) )
P=\(\frac{3}{\left(x^2-1\right)\left(x^2-x+1\right)}-\frac{1}{\left(x^2-1\right)\left(x^2+x+1\right)}-\frac{4}{\left(x^2-1\right)\left(x^2+x+1\right)\left(x^2-x+1\right)}\)
=\(\frac{2}{x^4+x^2+1}>0\)
P\(< \frac{32}{9}\Leftrightarrow\frac{2}{x^4+x^2+1}< \frac{32}{9}\)
\(\Leftrightarrow16x^4+16x^2+7>0\)
\(\Rightarrow\)\(0< P< \frac{32}{9}\) VOI X KHAC 1;-1
\(A=\left(\frac{\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}+\frac{1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right).\left(\frac{\sqrt{x}-2}{\sqrt{x}-1}\right)\)
\(=\frac{\left(x-2\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}.\frac{\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)}\)
\(=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}+2}\)
Để \(A< \frac{2}{3}\Rightarrow\frac{\sqrt{x}-1}{\sqrt{x}+2}< \frac{2}{3}\)
\(\Rightarrow3\sqrt{x}-3< 2\sqrt{x}+4\) (do \(\sqrt{x}+2>0\) \(\forall x\) xác định)
\(\Rightarrow\sqrt{x}< 7\Rightarrow x< 49\)
Kết hợp ĐKXĐ \(\Rightarrow\left\{{}\begin{matrix}0\le x< 49\\x\ne\left\{4;9\right\}\end{matrix}\right.\)
a) \(2\sqrt{x^2}=2.\left|x\right|=-2x\)(vì x<0)
b) \(\frac{1}{2}\sqrt{x^{10}}=\frac{1}{2}\sqrt{\left(x^5\right)^2}\frac{1}{2}\left|x^5\right|=-\frac{1}{2}x^5\)(vì x>0)
c) \(x-4+\sqrt{x^2-8x+16}=x-4+\sqrt{\left(x-4\right)^2}=x-4+\left|x-4\right|=x-4+4-x=0\)(vì x<4 nên x-4<0)
d) \(\frac{3-\sqrt{x}}{x-9}=\frac{-\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\frac{-1}{\sqrt{x}+3}\)