If frac{x^4+x^3+2x^2-2x+1}{x^3+x^2} = P(x) + | vedclass

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(C) Perform polynomial division on the expression: $\frac{x^4+x^3+2x^2-2x+1}{x^3+x^2} = x + \frac{2x^2-2x+1}{x^2(x+1)}$.Comparing this with $P(x) + \f

Vedclass · Accepted Answer

$3$

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(C) Perform polynomial division on the expression: $\frac{x^4+x^3+2x^2-2x+1}{x^3+x^2} = x + \frac{2x^2-2x+1}{x^2(x+1)}$.Comparing this with $P(x) + \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$,we have $P(x) = x$ and $\frac{2x^2-2x+1}{x^2(x+1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$.Multiplying by $x^2(x+1)$,we get $2x^2-2x+1 = Ax(x+1) + B(x+1) + Cx^2$.Setting $x = 0$,we get $1 = B(1) \Rightarrow B = 1$.Setting $x = -1$,we get $2(-1)^2 - 2(-1) + 1 = C(-1)^2$ $\Rightarrow 2+2+1 = C$ $\Rightarrow C = 5$.Comparing the coefficients of $x^2$: $2 = A + C$ $\Rightarrow 2 = A + 5$ $\Rightarrow A = -3$.Thus,$A+B+C = -3 + 1 + 5 = 3$.

If $\frac{x^4+x^3+2x^2-2x+1}{x^3+x^2} = P(x) + \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$,then $A+B+C = $

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