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Ta có:\(\frac{a}{\left(x+1\right)^3}+\frac{b}{\left(x+1\right)^2}=\frac{a+bx+b}{\left(x+1\right)^3}\)
Vì \(\frac{a+bx+b}{\left(x+1\right)^3}\) và \(\frac{3x+1}{\left(x+1\right)^3}\) đều có chung tử
Suy ra a+bx+b=3x+1
\(\frac{3x+1}{\left(x+1\right)^3}=\frac{a}{\left(x+1\right)^3}+\frac{b}{\left(x+1\right)^2}\Leftrightarrow\frac{3x+1}{\left(x+1\right)^3}=\frac{a}{\left(x+1\right)^3}+\frac{b.\left(x+1\right)}{\left(x+1\right)^3}\)
\(\Rightarrow\frac{3x+1}{\left(x+1\right)^3}-\frac{a+b.\left(x+1\right)}{\left(x+1\right)^3}=0\)\(\Rightarrow3x+1=a+b.\left(x+1\right)\)
Mà 3x+1=3.(x+1) -2 \(\Rightarrow b=3,a=-2\)
\(\frac{3x+1}{\left(x+1\right)^3}=\frac{a}{\left(x+1\right)^3}+\frac{b.\left(x+1\right)}{\left(x+1\right)^3}=\frac{bx+a+b}{\left(x+1\right)^3}\)
=>b = 3
=> a+b =1=> a= 1-b=1-3=-2
Vậy a = -2 ; b = 3
\(\frac{3x+1}{\left(x+1\right)^3}=\frac{a}{\left(x+1\right)^3}+\frac{b\left(x+1\right)}{\left(x+1\right)^3}\)
\(\Leftrightarrow\frac{3x+1}{\left(x+1\right)^3}=\frac{bx+a+b}{\left(x+1\right)^3}\)
Đồng nhất 2 vế ta được: \(\left\{{}\begin{matrix}b=3\\a+b=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=-2\\b=3\end{matrix}\right.\)
a, ĐKXĐ: \(\hept{\begin{cases}x^3+1\ne0\\x^9+x^7-3x^2-3\ne0\\x^2+1\ne0\end{cases}}\)
b, \(Q=\left[\left(x^4-x+\frac{x-3}{x^3+1}\right).\frac{\left(x^3-2x^2+2x-1\right)\left(x+1\right)}{x^9+x^7-3x^2-3}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\left[\frac{\left(x^3+1\right)\left(x^4-x\right)+x-3}{\left(x+1\right)\left(x^2-x+1\right)}.\frac{\left(x-1\right)\left(x+1\right)\left(x^2-x+1\right)}{\left(x^7-3\right)\left(x^2+1\right)}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\left[\left(x^7-3\right).\frac{\left(x-1\right)}{\left(x^7-3\right)\left(x^2+1\right)}+1-\frac{2\left(x+6\right)}{x^2+1}\right]\)
\(Q=\frac{x-1+x^2+1-2x-12}{x^2+1}\)
\(Q=\frac{\left(x-4\right)\left(x+3\right)}{x^2+1}\)
Xét vế phải : \(\frac{a}{x+1}+\frac{b}{x-2}+\frac{c}{\left(x-2\right)^2}=\frac{a\left(x-2\right)^2}{\left(x+1\right)\left(x-2\right)^2}+\frac{b\left(x-2\right)\left(x+1\right)}{\left(x+1\right)\left(x-2\right)^2}+\frac{c\left(x+1\right)}{\left(x+1\right)\left(x-2\right)^2}\)
\(=\frac{a\left(x^2-4x+4\right)+b\left(x^2-x-2\right)+c\left(x+1\right)}{\left(x+1\right)\left(x-2\right)^2}\)
\(=\frac{x^2\left(a+b\right)+x\left(-4a-b+c\right)+\left(4a-2b+c\right)}{\left(x+1\right)\left(x-2\right)^2}\)
So sánh với vế trái, suy ra :
\(\begin{cases}a+b=2\\-4a-b+c=-1\\4a-2b+c=1\end{cases}\). Giải ra được \(\left(a,b,c\right)=\left(\frac{4}{9};\frac{14}{9};\frac{7}{3}\right)\)
Ta có:
\(\frac{3x+1}{\left(x+1\right)^3}=\frac{a}{\left(x+1\right)^3}+\frac{b}{\left(x+1\right)^2}=\frac{bx+b+a}{\left(x+1\right)^3}\)
Đồng nhất thức 2 vế được: \(\hept{\begin{cases}b=3\\a+b=1\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}a=-2\\b=3\end{cases}}\)