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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.

Linear Factors

If the following is an identity in \(x\): \[\frac{A}{(x-2)(5x-8)}=\frac{1}{x-2}-\frac{B}{5x-8},\] what is the value of \(A+B?\)

How many terms would there be in the partial fraction decomposition of

\[ \frac{1}{ ( x- 5) ( x + 5) ( x + 9 )}? \]

If the following is an identity in \(x\):

\[ \frac{11x+50 }{(x+2)(x+6)} = \frac{A}{x+2} + \frac{B}{x+6}, \]

what is the value of \( A + B? \)

Which of the following is the correct partial fraction decomposition of

\[ \frac{ x+1 } { ( x + 2) ( x + 3) } ? \]

If the following is an identity in \(x\): \[\frac{5x-6}{x^2-3x+2}=\frac{a}{x-1}+\frac{b}{x-2},\] what is the value of \(a \times b\)?

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