If frac{(x - a)(x - b)}{(x - c)(x - d)} = fra | vedclass

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(A) Given the expression $\frac{(x - a)(x - b)}{(x - c)(x - d)} = \frac{A}{x - c} - \frac{B}{x - d} + C$. To find $C$,we perform polynomial long divis

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$1$

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(A) Given the expression $\frac{(x - a)(x - b)}{(x - c)(x - d)} = \frac{A}{x - c} - \frac{B}{x - d} + C$. To find $C$,we perform polynomial long division on the left side. The numerator is $(x - a)(x - b) = x^2 - (a + b)x + ab$. The denominator is $(x - c)(x - d) = x^2 - (c + d)x + cd$. Dividing $x^2 - (a + b)x + ab$ by $x^2 - (c + d)x + cd$: $\frac{x^2 - (a + b)x + ab}{x^2 - (c + d)x + cd} = 1 + \frac{(c + d - a - b)x + (ab - cd)}{x^2 - (c + d)x + cd}$. Comparing this with the given form $\frac{A}{x - c} - \frac{B}{x - d} + C$,we identify $C = 1$.

If $\frac{(x - a)(x - b)}{(x - c)(x - d)} = \frac{A}{x - c} - \frac{B}{x - d} + C$,then $C = $

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