Q. 29

Question

In Problems 13–46, write the partial fraction decomposition of each rational expression.

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

Step-by-Step Solution

Verified

Answer

The partial fraction decomposition of a rational expression is

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

1Step 1. Given information

Given rational expression is

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

2Step 2. Partial fraction decomposition

partial fraction decomposition of a rational expression

$\frac{P (x)}{Q (x)} = \frac{A_{1}}{x - a_{1}} + \frac{A_{2}}{x - a_{2}} + \dots + \frac{A_{n}}{x - a_{n}} \frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{A}{x + 1} + \frac{B x + C}{(x^{2} + 2 x + 4)} \dots (i) \frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{A (x^{2} + 2 x + 4)}{(x + 1) (x^{2} + 2 x + 4)} + \frac{(B x + C) (x + 1)}{(x + 1) (x^{2} + 2 x + 4)} x^{2} + 2 x + 3 = A (x^{2} + 2 x + 4) + (B x + C) (x + 1) x^{2} + 2 x + 3 = A x^{2} + 2 A x + 4 A + B x^{2} + B x + C x + C x^{2} + 2 x + 3 = (A + B) x^{2} + (2 A + B + C) x + (4 A + C) \dots (i i)$

3Step 3. Values of numerator coefficients and constants

Compare the coefficient of $x^{2}$ in equation ii

$A + B = 1 B = 1 - A$

Compare the constants in equation ii

$3 = 4 A + C C = 3 - 4 A$

Compare the coefficient of x in equation ii and Substitute the expression for B and C

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

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

4Step 4. partial fraction decomposition of a rational expression

Substitute the value of A, B, and C in the equation i

$\frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{A}{x + 1} + \frac{B x + C}{(x^{2} + 2 x + 4)} \frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{\frac{2}{3}}{x + 1} + \frac{\frac{1}{3} x + \frac{1}{3}}{(x^{2} + 2 x + 4)} \frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{\frac{2}{3}}{x + 1} + \frac{\frac{1}{3} (x + 1)}{(x^{2} + 2 x + 4)}$

So the partial fraction decomposition of a rational expression is $\frac{x^{2} + 2 x + 3}{(x + 1) (x^{2} + 2 x + 4)} = \frac{\frac{2}{3}}{x + 1} + \frac{\frac{1}{3} (x + 1)}{(x^{2} + 2 x + 4)}$

Q. 28

Q. 30

Other exercises in this chapter

Q. 27

Write the partial fraction decomposition of each rational expression.x+4x2(x2+4)

View solution

Q. 28

In Problems 13–46, write the partial fraction decomposition of each rational expression.10x2+2x(x-1)2(x2+2)

View solution

Q. 30

In Problems 13–46, write the partial fraction decomposition of each rational expression.x2-11x-18x(x2+3x+3)

View solution

Q. 31

In Problems 13–46, write the partial fraction decomposition of each rational expression.x(3x-2)(2x+1)

View solution