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Partial Fractions

Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.

Repeated Factors

         

If the following is an identity in \(x\): \[\frac{4x^2+27x+33}{(x-1)(x+3)^2}=\frac{a}{x-1}+\frac{bx+c}{(x+3)^2},\] what is the value of \(a+b+c\)?

Suppose \(a\), \(b\), and \(c\) are constants such that the following holds for all real numbers \(x\) such that all of the denominators are nonzero:

\[\frac{9}{x(x+1)^2}=\frac{a}{x}+\frac{b}{x+1}+\frac{c}{(x+1)^2}.\] What is the value of \(abc?\)

If the following is an identity in \(x\): \[\frac{17x^3+7x^2-17x-7}{(x-1)^2(x+1)^2}=\frac{A}{x-1}+\frac{B}{x+1},\] what is the value of \(A \times B?\)

If the following is an identity in \(x\): \[\frac{11x^2+90x+99}{(x-3)^2(x+3)^2}=\frac{A}{(x-3)^2}-\frac{B}{(x+3)^2},\] what is the value of \(A \times B?\)

If the following is an identity in \(x\): \[\frac{-13x+32}{(x+1)(x-2)^2}=\frac{a}{x+1}+ \frac{ b} { (x-2) } + \frac{ c} { (x-2)^2} , \] what is the value of \(a+b+c\)?

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