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Chapter 5: Problem 26

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{2}+49}{x(x+7)^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of \( \frac{3 x^{2}+49}{x(x+7)^{2}} \) is \( \frac{1}{x} - \frac{3}{x + 7} \)

Step by step solution

01

Identify the factors in the denominator

In this case, the denominator of the rational expression is \(x(x+7)^{2}\). This is composed of two parts, \(x\) and \((x+7)^{2}\). So we will need to decompose the rational expression into two fractions.
02

Set up the general form of the partial fraction decomposition

The general form of the partial fraction decomposition based on the factors identified in Step 1 is: \( \frac{3 x^{2}+49}{x(x+7)^{2}} = \frac{A}{x} + \frac{B}{x + 7} + \frac{C}{(x+7)^{2}} \), where A, B, and C are constants we need to determine.
03

Solve for A, B, and C

To solve for A, B, and C, we can first multiply both sides by the denominator of the left side: \( 3x^{2}+49 = Ax(x+7)^2 + Bx(x+7) + Cx = Ax^3 + 15Ax^2 + 49Ax + Bx^2 + 7Bx + Cx. Comparing the coefficients on each side of the equation, we have \(3A + B + C = 3\), \(15A + B + 7B = 0\), and \(49A = 49\). Solving this system of equations, we find \(A = 1\), \(B = -3\), and \(C = 0\)
04

Write the final partial fraction decomposition

Substituting \(A = 1\), \(B = -3\), and \(C = 0\) into the general form obtained in Step 2, we get the final partial fraction decomposition: \( \frac{3 x^{2}+49}{x(x+7)^{2}} = \frac{1}{x} - \frac{3}{x + 7} \

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