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a: ĐKXĐ: \(x\notin\left\{0;1;2;3;4;5\right\}\)
b: \(P=\dfrac{1}{x^2-x}+\dfrac{1}{x^2-3x+2}+\dfrac{1}{x^2-5x+6}+\dfrac{1}{x^2-7x+12}+\dfrac{1}{x^2-9x+20}\)
\(=\dfrac{1}{x\left(x-1\right)}+\dfrac{1}{\left(x-1\right)\left(x-2\right)}+\dfrac{1}{\left(x-2\right)\left(x-3\right)}+\dfrac{1}{\left(x-3\right)\left(x-4\right)}+\dfrac{1}{\left(x-4\right)\left(x-5\right)}\)
\(=\dfrac{-1}{x}+\dfrac{1}{x-1}-\dfrac{1}{x-1}+\dfrac{1}{x-2}-\dfrac{1}{x-2}+\dfrac{1}{x-3}-\dfrac{1}{x-3}+\dfrac{1}{x-4}-\dfrac{1}{x-4}+\dfrac{1}{x-5}\)
\(=\dfrac{1}{x-5}-\dfrac{1}{x}\)
\(=\dfrac{x-\left(x-5\right)}{x\left(x-5\right)}=\dfrac{5}{x\left(x-5\right)}\)
c: \(x^3-x^2+2=0\)
=>\(x^3+x^2-2x^2+2=0\)
=>\(x^2\cdot\left(x+1\right)-2\left(x-1\right)\left(x+1\right)=0\)
=>\(\left(x+1\right)\left(x^2-2x+2\right)=0\)
=>x+1=0
=>x=-1
Khi x=-1 thì \(P=\dfrac{5}{\left(-1\right)\left(-1-5\right)}=\dfrac{5}{\left(-1\right)\cdot\left(-6\right)}=\dfrac{5}{6}\)
Từ \(x\left(\dfrac{1}{y}+\dfrac{1}{z}\right)+y\left(\dfrac{1}{z}+\dfrac{1}{x}\right)+z\left(\dfrac{1}{x}+\dfrac{1}{y}\right)=-2\) ta có:
\(x^2y+y^2z+z^2x+xy^2+yz^2+zx^2+2xyz=0\)
\(\Leftrightarrow\left(x+y\right)\left(y+z\right)\left(z+x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+y=0\\y+z=0\\z+x=0\end{matrix}\right.\).
Không mất tính tổng quát, giả sử x + y = 0
\(\Leftrightarrow x=-y\)
\(\Leftrightarrow x^3=-y^3\).
Kết hợp với \(x^3+y^3+z^3=1\) ta có \(z^3=1\Leftrightarrow z=1\).
Vậy \(P=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{-y}+\dfrac{1}{y}+\dfrac{1}{1}=1\).
\(6xy=x+y\ge2\sqrt[]{xy}\Rightarrow\sqrt{xy}\ge\dfrac{1}{3}\Rightarrow xy\ge\dfrac{1}{9}\Rightarrow\dfrac{1}{xy}\le9\)
\(M=\dfrac{\dfrac{x+1}{xy+1}+\dfrac{xy+x}{1-xy}+1}{1+\dfrac{xy+x}{1-xy}-\dfrac{x+1}{xy+1}}=\dfrac{\dfrac{x+1}{xy+1}+\dfrac{x+1}{1-xy}}{\dfrac{x+1}{1-xy}-\dfrac{x+1}{xy+1}}=\dfrac{\dfrac{1}{1-xy}+\dfrac{1}{1+xy}}{\dfrac{1}{1-xy}-\dfrac{1}{1+xy}}\)
\(M=\dfrac{1+xy+1-xy}{1+xy-1+xy}=\dfrac{2}{2xy}=\dfrac{1}{xy}\le9\)
Dấu "=" xảy ra khi \(x=y=\dfrac{1}{3}\)
a) A = \(\dfrac{1}{x-1}-\dfrac{4}{x+1}+\dfrac{8x}{\left(x-1\right)\left(x+1\right)}\)
= \(\dfrac{x+1-4x+4+8x}{\left(x-1\right)\left(x+1\right)}=\dfrac{5x+5}{\left(x-1\right)\left(x+1\right)}=\dfrac{5}{x-1}\) => đpcm
b) \(\left|x-2\right|=3=>\left[{}\begin{matrix}x-2=3< =>x=5\left(C\right)\\x-2=-3< =>x=-1\left(L\right)\end{matrix}\right.\)
Thay x = 5 vào A, ta có:
A = \(\dfrac{5}{5-1}=\dfrac{5}{4}\)
c) Để A nguyên <=> \(5⋮x-1\)
| x-1 | -5 | -1 | 1 | 5 |
| x | -4(C) | 0(C) | 2(C) | 6(C) |
`a/(x+1)+b/(x-2)=(a(x-2)+b(x+1))/((x+1)(x-2))`
`=(ax-2a+bx+b)/(x^2-x-2)`
`=((a+b)x+(-2a+b))/(x^2-x-2)`
``
Theo đề bài: `((a+b)x+(-2a+b))/(x^2-x-2)=(32x-19)/(x^2-x-2)`
Đồng nhất hệ số ta được: `{(a+b=32),(-2a+b=-19):}`
`<=>{(a+b=32),(2a-b=19):}`
`<=>{(3a=51),(a+b=32):}`
`<=>{(a=17),(17+b=32):}`
`<=>{(a=17),(b=15):}`
\(1+\dfrac{1}{x+2}=\dfrac{12}{x^3+8}\Leftrightarrow\dfrac{\left(x^3+8\right)\left(x+2\right)}{\left(x^3+8\right)\left(x+2\right)}+\dfrac{\left(x^3+8\right)}{\left(x^3+8\right)\left(x+2\right)}=\dfrac{12\left(x+2\right)}{\left(x^3+8\right)\left(x+2\right)}\)
\(\Rightarrow x^4+2x^3+8x+16+x^3+8=12x+24\)
\(\Leftrightarrow x^4+3x^3-4x=0\\ \Leftrightarrow x\left(x^3+3x^2-4\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\left(loại\right)\\x^3+3x^2-4=0\end{matrix}\right.\)
\(x^3+3x^2-4=0\Leftrightarrow\left(x^3+4x^2+4x\right)-\left(x^2+4x+4 \right)=0\)
\(\left(x-1\right)\left(x^2+4x+4\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x-1=0\\x^2+4x+4=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=1\\\left(x+2\right)^2=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=1\\x=-2\left(loại\right)\end{matrix}\right.\)
vậy phương trình có tập nghiệm là S={1}