Cho \(P=\frac{x^3+4x^2-17x-60}{x^3-4x^2+7x-28}\)
a, Rút gọn P
b, Với giá trị nào của x thì P≥0
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D
27 tháng 6 2018
\(a,\)
\(A=\left(\frac{4x}{x+2}-\frac{x^3-8}{x^3+8}.\frac{4x^2-4x+16}{x^2-4}\right):\frac{16}{x+2}.\frac{x^2+3x+2}{x^2+x+1}\)\(ĐKXĐ:x\ne\pm2\)
\(A=[\frac{4x}{x+2}-\frac{\left(x-2\right)\left(x^2+2x+4\right).4\left(x^2-2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)\left(x-2\right)\left(x+2\right)}]:\frac{16}{x+2}.\frac{\left(x+1\right)\left(x+2\right)}{x^2+x+1}\)
\(A=[\frac{4x}{x+2}-\frac{4\left(x^2+2x+4\right)}{\left(x+2\right)^2}].\frac{x+2}{16}.\frac{\left(x+1\right)\left(x+2\right)}{x^2+x+1}\)
\(A=\frac{4x^2+8x-4x^2-8x-16}{\left(x+2\right)^2}.\frac{x+2}{16}.\frac{\left(x+1\right)\left(x+2\right)}{x^2+x+1}\)
\(A=\frac{16\left(x+2\right)}{\left(x+2\right)^2.16}.\frac{\left(x+1\right)\left(x+2\right)}{x^2+x+1}\)
\(A=\frac{-\left(x+1\right)}{x^2+x+1}\)
\(B=\frac{x^2+x-2}{x^3-1}\)\(ĐKXĐ:x\ne1\)
\(B=\frac{\left(x+2\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(B=\frac{x+2}{x^2+x+1}\)
\(b,\)
Ta có:
\(A+B=\frac{-\left(x+1\right)}{x^2+x+1}+\frac{x+2}{x^2+x+1}\)
\(=\frac{-x-1+x+2}{x^2+x+1}\)
\(=\frac{1}{x^2+x+1}\)
\(\Rightarrow A+B=\frac{1}{x^2+x+1}=\frac{1}{x^2+2.x.\left(\frac{1}{2}\right)^2+\frac{3}{4}}=\frac{1}{\left(x+\frac{1}{2}\right)^2}+\frac{3}{4}\)
Vì:\(\left(x+\frac{1}{2}\right)^2\ge0\forall x\)
\(\Rightarrow\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\forall x\)
\(\Rightarrow\frac{1}{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}\le\frac{1}{\frac{3}{4}}\)
\(\Rightarrow A+B\le\frac{4}{3}\)
\(\Rightarrow GTLN\)của \(A+B=\frac{4}{3}\Leftrightarrow x+\frac{1}{2}=0\)
\(\Leftrightarrow x=\frac{-1}{2}\left(TMĐK\right)\)
Vậy........
MN
3 tháng 3 2020
\(ĐKXĐ:\hept{\begin{cases}x\ne\pm2\\x\ne0\end{cases}}\)
a) \(P=\left(\frac{x^2}{x^3-4x}+\frac{6}{6-3x}+\frac{1}{x+2}\right):\left(x-2+\frac{10-x^2}{x+2}\right)\)
\(\Leftrightarrow P=\left(\frac{x^2}{x\left(x-2\right)\left(x+2\right)}-\frac{6}{3\left(x-2\right)}+\frac{1}{x+2}\right):\frac{x^2-4+10-x^2}{x-2}\)
\(\Leftrightarrow P=\frac{x^2-2x\left(x+2\right)+x\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}:\frac{6}{x-2}\)
\(\Leftrightarrow P=\frac{x^2-2x^2-4x+x^2-2x}{x\left(x-2\right)\left(x+2\right)}\cdot\frac{x-2}{6}\)
\(\Leftrightarrow P=\frac{-6x}{6x\left(x+2\right)}\)
\(\Leftrightarrow P=\frac{-1}{x+2}\)
b) Khi \(\left|x\right|=\frac{3}{4}\)
\(\Leftrightarrow\orbr{\begin{cases}x=\frac{3}{4}\\x=-\frac{3}{4}\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}P=-\frac{1}{\frac{3}{4}+2}=-\frac{4}{11}\\P=-\frac{1}{-\frac{3}{4}+2}=-\frac{4}{5}\end{cases}}\)
c) Để P = 7
\(\Leftrightarrow-\frac{1}{x+2}=7\)
\(\Leftrightarrow7\left(x+2\right)=-1\)
\(\Leftrightarrow7x+14=-1\)
\(\Leftrightarrow7x=-15\)
\(\Leftrightarrow x=-\frac{15}{7}\)
Vậy để \(P=7\Leftrightarrow x=-\frac{15}{7}\)
d) Để \(P\inℤ\)
\(\Leftrightarrow1⋮x+2\)
\(\Leftrightarrow x+2\inƯ\left(1\right)=\left\{\pm1\right\}\)
\(\Leftrightarrow x\in\left\{-3;-1\right\}\)
Vậy để \(P\inℤ\Leftrightarrow x\in\left\{-3;-1\right\}\)
6 tháng 2 2019
a) ĐKXĐ : \(x\ne\pm1\)
+ \(B=\left(\frac{\left(x+1\right)^2-\left(x-1\right)^2+\left(x^2-4x-1\right)}{x^2-1}\right)\cdot\frac{x-2014}{x-1}\)
\(B=\frac{4x+x^2-4x-1}{x^2-1}\cdot\frac{x-2014}{x+1}\)
\(B=\frac{x^2-1}{x^2-1}\cdot\frac{x-2014}{x+1}=\frac{x-2014}{x+1}\)\
b) B có giá trị nguyên
\(\Leftrightarrow x-2014⋮x+1\)
\(\Leftrightarrow x+1-2015⋮x+1\)
\(\Leftrightarrow2015⋮x+1\)
21 tháng 7 2016
a) ĐKXĐ: \(x\ne\left\{-3;-\frac{1}{3}\right\}\)
Ta có: \(\frac{3x-1}{3x+1}+\frac{x-3}{x+3}=\)\(\frac{\left(3x-1\right)\left(x+3\right)+\left(x-3\right)\left(3x+1\right)}{\left(3x+1\right)\left(x+3\right)}\)=\(\frac{3x^2+9x-x-3+3x^2+x-9x-3}{3x^2+9x+x+3}\)
= \(\frac{6x^2-6}{3x^2+10x+3}\)
=> \(\frac{6x^2-6}{3x^2+10x+3}=2\)
<=> \(6x^2-6=6x^2+20x+6\)
<=> 20x=12
<=>x=\(\frac{12}{20}=\frac{3}{5}\)
Vậy x=3/5
\(a,ĐKXĐ:x\ne4\\ P=\frac{x^3+4x^2-17x-60}{x^3-4x^2+7x-28}\\ =\frac{x^3-4x^2+8x^2-32x+15x-60}{x^2\left(x-4\right)+7\left(x-4\right)}\\ =\frac{x^2\left(x-4\right)+8x\left(x-4\right)+15\left(x-4\right)}{\left(x-4\right)\left(x^2+7\right)}\\ =\frac{\left(x-4\right)\left(x^2+3x+5x+15\right)}{\left(x-4\right)\left(x^2+7\right)}\\ =\frac{\left(x+3\right)\left(x+5\right)}{x^2+7}\)
b, Với x khác 4, ta có:
làm câu b giúp mình đi