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Calculus / Calculus: Early Transcendentals 1 / Chapter 7.4 / Problem 5E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Chapter 4, Problem 5E

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QUESTION:

Setting up partial fraction decomposition Give the appropriate form of the partial fraction decomposition for the following functions.

\(\frac{2}{x^{2}-2 x-8}\)

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QUESTION:

Setting up partial fraction decomposition Give the appropriate form of the partial fraction decomposition for the following functions.

\(\frac{2}{x^{2}-2 x-8}\)

ANSWER:

Problem 5E

Give the appropriate form of the partial fraction decomposition for the following functions.

$\frac{2}{{x}^{2}-2x-8}$

Solution:-
Step 1

To obtain the partial fraction decomposition for the following function, first we arrange the fraction as follows

$\frac{2}{{x}^{2}-2x-8}=\frac{2}{{x}^{2}-4x+2x-8}$

= $\frac{2}{x\left({x-4}\right)+2\left({x-4}\right)}$

= $\frac{2}{\left({x+2}\right)\left({x-4}\right)}$
Now, decomposing the fraction $\frac{2}{\left({x+2}\right)\left({x-4}\right)}$ into partial fractions.
$\frac{2}{\left({x+2}\right)\left({x-4}\right)}=\ \frac{A}{x+2}+\frac{B}{x-4}$
= $\frac{A\left({x-4}\right)+B\left({x+2}\right)}{\left({x+2}\right)\left({x-4}\right)}$
= $\frac{Ax\ +\ Bx\ -4A+2B}{\left({x+2}\right)\left({x-4}\right)}$
= $\frac{x\left({A+B}\right)\ \ -4A+2B}{\left({x+2}\right)\left({x-4}\right)}$

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