Find the partial fraction decomposition for (2x2 - 10x + 2)/(x - 2)(x2 + 2x + 2).

To find the partial fraction decomposition for 2x2-10x+2x-2x2+2x+2. Solution: Decomposing the given…

Answered: Find the partial fraction decomposition…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.10: Partial Fractions

Problem 12E

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Question

Find the partial fraction decomposition for (2x² - 10x + 2)/(x - 2)(x² + 2x + 2).

Expert Solution

Step 1

To find the partial fraction decomposition for $\frac{(2 x^{2} - 10 x + 2)}{(x - 2) (x^{2} + 2 x + 2)}$ .

Solution:

Decomposing the given expression using partial fraction.

$\begin{array}{rcl} \frac{(2 x^{2} - 10 x + 2)}{(x - 2) (x^{2} + 2 x + 2)} & = & \frac{A}{(x - 2)} + \frac{B x + C}{(x^{2} + 2 x + 2)} \\ \frac{(2 x^{2} - 10 x + 2)}{(x - 2) (x^{2} + 2 x + 2)} & = & \frac{(x^{2} + 2 x + 2) A + (x - 2) (B x + C)}{(x - 2) (x^{2} + 2 x + 2)} \\ (2 x^{2} - 10 x + 2) & = & A x^{2} + 2 A x + 2 A + B x^{2} + C x - 2 B x - 2 C \\ 2 x^{2} - 10 x + 2 & = & (A + B) x^{2} + (2 A - 2 B + C) x + (2 A - 2 C) \end{array}$

Step 2

On comparing both sides of the equation, we get

$\begin{array}{rcl} A + B & = & 2 \dots (1) \\ 2 A - 2 B + C & = & - 10 \dots (2) \\ 2 A - 2 C & = & 2 \dots (3) \end{array}$

From equation (1),

$B = 2 - A$

Put $B = 2 - A$ in equation (2),

$\begin{array}{rcl} 2 A - 2 (2 - A) + C & = & - 10 \\ 2 A - 4 + 2 A & = & - 10 \\ 4 A & = & - 10 + 4 \\ 4 A & = & - 6 \\ A & = & - \frac{6}{4} \\ A & = & - \frac{3}{2} \end{array}$

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