Chapter 11: Q. 20 (page 765)

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

Short Answer

Expert verified

The partial fraction decomposition is $\frac{- 3}{4 x} + \frac{- 1}{2 x^{2}} + \frac{3}{4 (x - 2)}$ .

Step by step solution

Step 1. Given Information.

The rational expression is,

$\frac{x + 1}{x^{2} (x - 2)}$ .

Step 2. Partial fraction decomposition.

Decomposition of the partial fraction,

$\frac{x + 1}{x^{2} (x - 2)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 2} . . . . . . . . . . . . (1)$

Multiplying both sides by $x^{2} (x - 2)$ ,

$x + 1 = A x (x - 2) + B (x - 2) + C x^{2} x + 1 = A (x^{2} - 2 x) + B (x - 2) + C x^{2} x + 1 = (A + C) x^{2} + (- 2 A + B) x - 2 B . . . . . . . (2)$

Equating the coefficients of the like powers of $x$ , we get,

$A + C = 0 . . . . . . (3) - 2 A + B = 1 . . . . . . . . (4) - 2 B = 1 . . . . . . . . (5)$

Solving the equations we get,

$B = - \frac{1}{2} A = - \frac{3}{4} C = \frac{3}{4}$

The partial fraction decomposition is,

$\frac{x + 1}{x^{2} (x - 2)} = \frac{- 3}{4 x} + \frac{- 1}{2 x^{2}} + \frac{3}{4 (x - 2)}$

Step 3. Checking the solution.

Adding the rational expressions,

$\frac{- 3}{4 x} + \frac{- 1}{2 x^{2}} + \frac{3}{4 (x - 2)} = \frac{- 3 x (x - 2) + (- 1) (2 (x - 2)) + 3 x^{2}}{4 x^{2} (x - 2)}$

$= \frac{- 3 (x^{2} - 2 x) + (- 1) (2 x - 4) + 3 x^{2}}{4 x^{2} (x - 2)} = \frac{- 3 x^{2} + 6 x - 2 x + 4 + 3 x^{2}}{4 x^{2} (x - 2)} = \frac{4 x + 4}{4 x^{2} (x - 2)} = \frac{4 (x + 1)}{4 x^{2} (x - 2)} = \frac{x + 1}{x^{2} (x - 2)}$

The solution is true.

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial fraction decomposition.

Step 3. Checking the solution.

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Write the partial fraction decomposition of each rational expression.x+1x2(x-2).

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Partial fraction decomposition.

Step 3. Checking the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

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