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Chapter 5: Problem 40

Write the partial fraction decomposition of each rational expression. $$\frac{3 x^{3}-6 x^{2}+7 x-2}{\left(x^{2}-2 x+2\right)^{2}}$$

Short Answer

Expert verified
Unfortunately, since the exercise did not include specific numerical coefficients for A, B, and C, it is not possible to provide a numerical answer. The answer would be a function in terms of x, expressed as A + B/(x^2-2x+2) + C/(x^2-2x+2)^2 with A, B, and C derived from the solutions to the equations.

Step by step solution

01

Polynomial Division

Since the degree of the numerator is more than the degree of the denominator, we first need to perform polynomial division of the numerator by the denominator. Here, we will use long division of polynomials. The goal here is to rewrite the expression in a way that the degree of the top polynomial is strictly less than the degree of the denominator.
02

Rewrite the rational expression

As a result from the long division we performed in Step 1, the obtained expression should be written in the form of \(A+B/(x^2-2x+2)+C/(x^2-2x+2)^2\), where A, B, and C are expressions to be determined. Here, A is the quotient of the division, and B and C are remaining parts of the numerator.
03

Solve for A, B and C

Now you need to equate this expression to the initial rational function. In doing so, you will obtain a system of equations on A, B, and C. By solving this system of equations, you can find the values of A, B and C.
04

Write the final decomposition

Finally, use the found values of A, B and C to present the final partial fractions decomposition of the initial rational function. Include this in the final answer.

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Most popular questions from this chapter

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(y=3 x-5\) \(21 x-35=7 y\)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x=3 y+7\\\ &x=2 y-1 \end{aligned} $$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=6\\\ &y=2 x \end{aligned} $$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=4\\\ &y=3 x \end{aligned} $$

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?
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