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Chapter 5: Problem 35

Write the partial fraction decomposition of each rational expression. $$\frac{6 x^{2}-x+1}{x^{3}+x^{2}+x+1}$$

Short Answer

Expert verified
The partial fraction decomposition of the rational expression is \(\frac{1}{x} + \frac{5x - 2}{x^{2}+x+1}\)

Step by step solution

01

Factorize the Denominator of the Rational Expression

Begin by factorizing the denominator of the given expression. Here, \(x^{3}+x^{2}+x+1 = x(x^{2}+x+1+1)\). However, \(x^{2}+x+1\) isn't factorisable using real numbers. So the factored form of the denominator is \(x(x^{2}+x+1)\).
02

Decompose into Partial Fraction

The next step is to decompose the fraction. We rewrite the rational function as the sum of partial fractions. Since the denominator was previously factored as \(x(x^{2}+x+1)\), the decomposition of the given rational expression can be expressed as: \(\frac{6 x^{2}-x+1}{x^{3}+x^{2}+x+1} = \frac{A}{x} + \frac{Bx+C}{x^{2}+x+1}\).
03

Equate Coefficients and Solve

Clear the fraction by multiplying both sides by \(x^{3}+x^{2}+x+1\), resulting in: \(6x^{2}-x+1 = A(x^{2}+x+1) + (Bx+C)x\). Compare the coefficients on both sides and solve for A, B, and C. Coefficients of \(x^{2}\) on left side: 6; on right side: A+B. So, A + B = 6. Coefficients of \(x\) on left side: -1; on right side: A + C. So, A + C = -1. Constants on left side: 1; on right side: A. So, A = 1. Substitute A = 1 into the other equations to get B = 5 and C = -2. The partial fraction decomposition of the given rational expression is therefore given by: \(\frac{1}{x} + \frac{5x - 2}{x^{2}+x+1}\)

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Most popular questions from this chapter

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$ y \leq 4 x+4 $$

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot, and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280-\) What are the lot's dimensions?

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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What is a system of linear equations? Provide an example with your description.
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