Chapter 11: Q. 17 (page 765)

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

Short Answer

Expert verified

The partial fraction decomposition is,

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

Step by step solution

Step 1. Given Information

The rational expression is,

\frac{x}{(x - 1) (x - 2)}

Step 2. Decompose the partial fractions.

Decompose the rational expressions,

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

Multiplying both sides by,

$x = A (x - 2) + B (x - 1) x = (A + B) x + (- 2 A - B) . . . . . . . . . . . (2)$

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

$A + B = 1 - 2 A - B = 0$

Solving the equations we get,

$A = - 1 B = 2$

The partial fraction decomposition is,

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

Step 3. Checking the solution.

Adding the rational expressions,

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

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Decompose the partial fractions.

Step 3. Checking the solution.

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Decompose the partial fractions.

Step 3. Checking the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Discrete Mathematics

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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