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Chapter 5: Problem 56

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

Short Answer

Expert verified
The decomposition of a rational expression with a prime quadratic factor in the denominator requires identification of the prime quadratic term, rewriting the expression as a sum of partial fractions, forming a polynomial equation by clearing fractions, comparing coefficients on both sides and solving the system of equations, finally substituting the computed values back to the partial fraction decomposition.

Step by step solution

01

Identify the Quadratic Factor

Given a rational expression, identify the prime quadratic factor in the denominator. A prime quadratic factor is a quadratic expression that cannot be factored any further.
02

Write the Original Expression as Equivalent Sum of Partial Fractions

Write the given rational expression equivalent to the partial fractions sum. The sum will be of the form \( A/(linear term) + B/(prime quadratic term) \) where A and B are numbers to be found. For the quadratic term, denominator factor has degree 2, so the numerator must be linear.
03

Clear the Fractions to Form a Polynomial Equation

Multiply each term of the equation by the common denominator of the fractions to clear the fractions and form a polynomial equation.
04

Compare the Coefficients

Match the coefficients of similar terms on both sides of the equation. This now becomes a system of linear equations.
05

Solve the System of Equations

Solve the system of linear equations to find the values of A and B. These values will be used in constructing the final partial fractions.
06

Write the Partial Fraction Decomposition

Substitute the values of A and B into the sum of partial fractions obtained initially to obtain the final partial fraction decomposition of the given rational expression.

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Most popular questions from this chapter

Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$ Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$ \frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2} $$

Find the partial fraction decomposition of $$ \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} $$

Explain what is meant by the partial fraction decomposition of a rational expression.

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

write the partial fraction decomposition of each rational expression. $$ \frac{3 x-5}{x^{3}-1} $$
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