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Chapter 5: Problem 25

Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}+2 x+7}{x(x-1)^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of \( \frac{x^{2}+2 x+7}{x(x-1)^{2}} \) is \( - \frac{7}{x} - \frac{2}{x-1} + \frac{19}{(x-1)^2} \)

Step by step solution

01

Identify the Form

Let's first write the rational expression in the form of partial fractions. We assume:\( \frac{x^{2}+2 x+7}{x(x-1)^{2}} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{(x-1)^2} \), \nwhere A, B, and C are constants that we need to find.
02

Clearing the fractions

To eliminate the fractional expression, we multiply each side of the equation by \(x(x-1)^{2}\), which gives us:\( x^2 + 2x + 7 = A(x-1)^2 + Bx(x-1) + Cx \)
03

Equating Coefficients

Now we put different values of x to get system of linear equations and find the constants A, B and C. Picking x = 0 and x = 1 simplifies the process.Put x = 0, => 7 = -APut x = 1, => 10 = A + B + C
04

Value for the second derivative

For the third equation, take the second derivative (x^2 + 2x + 7)'' and (A(x-1)虏+Bx(x-1)+Cx)'' and compare them at x = 1, we get: 0 = 2A + B.
05

Solving for A, B, and C

Solving these, we get A = -7, B = -2 and C = 19.
06

Writing the Partial Fraction Decomposition

Therefore, the partial fraction decomposition of the function is \( \frac{x^{2}+2 x+7}{x(x-1)^{2}} =- \frac{7}{x} - \frac{2}{x-1} + \frac{19}{(x-1)^2} \)

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