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Chapter 5: Problem 54

Explain how to find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator.

Short Answer

Expert verified
The partial fraction decomposition of a rational expression with distinct linear factors in the denominator involves factoring the denominator, setting up the partial fraction decomposition structure, equating it to the original fraction, solving for the unknowns, and then substitifying the resolved values back into the partial fraction structure.

Step by step solution

01

Factoring the denominator

Factoring the denominator of the rational expression or identify the distinct linear factors if the expression is already simplified. In general, any rational function will be in the form: \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials in \(x\). Factoring \(Q(x)\) will give distinct linear factors.
02

Writing the partial fraction decomposition

Setting up the partial fraction decomposition involves writing the expression \(\frac{P(x)}{Q(x)}\) as the sum of simpler fractions, where each denominator factor is in the denominator of one of these fractions. The form for a decomposition for distinct linear factors in the denominator will be: \(\frac{P(x)}{Q(x)} = \frac{A}{a} + \frac{B}{b} + \frac{C}{c}+ ... + \frac{N}{z}\), depending on how many factors you have.
03

Equating to Original Fraction

First, get rid of the fractions by multiplying through by the common denominator, \(Q(x)\): \( P(x) = AQ(x/a) + BQ(x/b) + CQ(x/c) + ... + NQ(x/z)\). Then, choose convenient values for \(x\) that will allow you to easily solve for the variables \(A,B,C,...,N\). These can usually be the roots of the linear factors in the denominator \(Q(x)\).
04

Solving for the unknowns

Solve for the variables \(A,B,C,...,N\) through substituting the chosen \(x\) values into the equation from Step 3.
05

Writing the Final Solution

Substitute the solved values of \(A, B, C, ..., N\) into partial fraction decomposition established in step 2. This will give the partial fraction decomposition of the rational expression.

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Most popular questions from this chapter

Consider the following array of numbers:$$\left[\begin{array}{ccc}1 & 2 & -1 \\\ 4 & -3 &-15\end{array}\right]$$ Rewrite the array as follows: Multiply each number in the top row by \(-4\) and add this product to the corresponding number in the bottom row. Do not change the numbers in the top row.

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$ \frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, \(x, y,\) and \(z\) -cannot contain an equation in the form \(y=m x+b\)

write the partial fraction decomposition of each rational expression. $$ \frac{3 x+50}{(x-9)(x+2)} $$

write the partial fraction decomposition of each rational expression. $$ \frac{3 x^{2}+49}{x(x+7)^{2}} $$
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