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Chapter 5: Problem 21

Write the partial fraction decomposition of each rational expression. $$\frac{6 x-11}{(x-1)^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of the given rational function is \(\frac{6x-11}{(x-1)^{2}} = \frac{16}{x-1} - \frac{5}{(x-1)^2}\).

Step by step solution

01

Setup of the Partial Fraction Decomposition

A rational function with denominator \((x-1)^2\) is expected to decompose to \[\frac{6x-11}{(x-1)^{2}} = \frac{A}{x-1} + \frac{B}{(x-1)^2}\] where A and B are constants that we would need to solve for.
02

Solving for Constants

Multiplying through by the divisor to eliminate the fractions: \[6x - 11 = A(x-1) + B\], we can now calculate the constants A and B. Let's put x=1 in the equation: we have \(B = 6 - 11 = -5\). Now, take the derivative of both sides of the equation: \(6 = A + 2B(x-1)\). Again let's put x=1: we get \(A = 6 - 2*(-5) = 16\).
03

The Partial Fraction Decomposition

With the values of A and B found, the partial fraction decomposition then becomes: \[\frac{6x-11}{(x-1)^{2}} = \frac{16}{x-1} - \frac{5}{(x-1)^2}\]

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