If frac{x^2+x+1}{x^2+2x+1}=A+frac{B}{x+1}+fra | vedclass

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(D) Given the equation: $\frac{x^2+x+1}{x^2+2x+1} = A + \frac{B}{x+1} + \frac{C}{(x+1)^2}$ Multiply both sides by $(x+1)^2$: $x^2+x+1 = A(x+1)^2 + B(x

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$2C$

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(D) Given the equation: $\frac{x^2+x+1}{x^2+2x+1} = A + \frac{B}{x+1} + \frac{C}{(x+1)^2}$ Multiply both sides by $(x+1)^2$: $x^2+x+1 = A(x+1)^2 + B(x+1) + C$ $x^2+x+1 = A(x^2+2x+1) + Bx + B + C$ $x^2+x+1 = Ax^2 + (2A+B)x + (A+B+C)$ Comparing the coefficients of $x^2$,$x$,and the constant term: $A = 1$ $2A+B = 1$ $\Rightarrow 2(1)+B = 1$ $\Rightarrow B = -1$ $A+B+C = 1$ $\Rightarrow 1-1+C = 1$ $\Rightarrow C = 1$ Now,calculate $A-B$: $A-B = 1 - (-1) = 2$ Since $C = 1$,we have $2 = 2C$. Therefore,$A-B = 2C$.

If $\frac{x^2+x+1}{x^2+2x+1}=A+\frac{B}{x+1}+\frac{C}{(x+1)^2}$,then $A-B$ is equal to

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