6x−1(x−3)(x+1)=Ax−3+Bx+1=A(x+1)+B(x−3)(x−3)(x+1)=Ax+A+Bx−3B(x−3)(x+1)=x(A+B)+A−3B(x−3)(x+1)\frac{6x-1}{(x-3)(x+1)}=\frac{A}{x-3}+\frac{B}{x+1}=\frac{A(x+1)+B(x-3)}{(x-3)(x+1)}=\frac{Ax+A+Bx-3B}{(x-3)(x+1)}=\frac{x(A+B)+A-3B}{(x-3)(x+1)}(x−3)(x+1)6x−1=x−3A+x+1B=(x−3)(x+1)A(x+1)+B(x−3)=(x−3)(x+1)Ax+A+Bx−3B=(x−3)(x+1)x(A+B)+A−3B
{A+B=6A−3B=−1\begin{cases} A+B=6 \\ A-3B=-1 \end{cases}{A+B=6A−3B=−1
{A+B=6A=3B−1\begin{cases} A+B=6 \\ A=3B-1 \end{cases}{A+B=6A=3B−1
3B−1+B=63B-1+B=63B−1+B=6
4B=74B=74B=7
B=74B=\frac{7}{4}B=47
A=3⋅74−1=174A=3\cdot\frac{7}{4}-1=\frac{17}{4}A=3⋅47−1=417
6x−1(x−3)(x+1)=174(x−3)+74(x+1)\frac{6x-1}{(x-3)(x+1)}=\frac{17}{4(x-3)}+\frac{7}{4(x+1)}(x−3)(x+1)6x−1=4(x−3)17+4(x+1)7
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