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Chapter 7: Problem 53

Write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{4 x^{2}-1}{2 x(x+1)^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of the given rational expression is \(\frac{-1}{2x} + \frac{5}{2(x+1)} + \frac{3}{2(x+1)^{2}}\).

Step by step solution

01

Identify the Factors of the Denominator

The denominator is \(2x(x+1)^{2}\), so we have two factors here: \(2x\) and \((x+1)^{2}\). We can write the general form of partial fractions as: \(\frac{A}{2x} + \frac{B}{x+1} + \frac{C}{(x+1)^{2}}\).
02

Generate Equations

Setting the given expression equal to the general form and clearing the fractions gives us an equation: \(4x^{2}-1 = A(x+1)^{2} + B*2x(x+1) + C*2x\). Expanding out gives \(4x^{2}-1 = A(x^{2} + 2x + 1) + 2Bx^{2} + 2Bx + 2Cx\). Grouping like terms gives \(4x^{2}-1 = (A+2B)x^{2} + (2A+2B+2C)x + A\). Now, matching coefficients, we have three equations: \(A + 2B = 4\), \(2A + 2B + 2C = 0\) and \(A = -1\). By solving these three equations, we can find the values for A, B and C.
03

Solve the System of Equations

From equation \(3\), we get: \(A=-1\). Substituting this value into equation \(1\), we get: \(B=\frac{5}{2}\). And substituting \(A\) and \(B\) into equation \(2\), we find that \(C=\frac{3}{2}\).
04

Substitute Back to Get the Partial Fractions

Substituting the values A, B, and C into \(\frac{A}{2x} + \frac{B}{x+1} + \frac{C}{(x+1)^{2}}\), we get \(\frac{-1}{2x} + \frac{5}{2(x+1)} + \frac{3}{2(x+1)^{2}}\).

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