Q. 13 In Exercises 12鈥�14, suppose th... [FREE SOLUTION]

Chapter 5: Q. 13 (page 441)

In Exercises 12鈥�14, suppose that you want to obtain a partialfraction decomposition of a rational function $\frac{p (x)}{q (x)}$ according to Theorem 5.12.
If q(x) is an irreducible quadratic, what can you say about its partial-fraction decomposition? What if q(x) is a reducible quadratic? Consider the functions $q (x) = x^{2} + 1 a n d q (x) = (x - 1) (x - 2)$ to find your answer.

Short Answer

Expert verified

If the function is irreducible, nothing further can be done but if function is reducible we get, $\frac{A_{1}}{l_{1} (x)} + \frac{A_{2}}{l_{2} (x)} where l_{1} (x) = x - 1 and l_{2} (x) = x - 2$

Step by step solution

Step 1. Given Information

The given functions are $q (x) = x^{2} + 1 a n d q (x) = (x - 1) (x - 2)$

Step 2. Explanation

If q(x) is an irreducible quadratic then $\frac{p (x)}{q (x)}$ is its own partial fractions decomposition, there is nothing further we can decompose.

If q(x) is a reducible quadratic then q(x) is a product of $l_{1} (x) and l_{2} (x)$ of two linear functions and obtain a partial decomposition of the form

$\frac{A_{1}}{l_{1} (x)} + \frac{A_{2}}{l_{2} (x)}$

As the given function is

$q (x) = x^{2} + 1, So, l_{1} (x) = x - 1 and l_{2} (x) = x - 2$

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Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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