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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 19
Chapter 6, Problem 19

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x2)/(x2 + 2x + 1)

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1
First, recognize that the denominator \(x^2 + 2x + 1\) can be factored. Factor it as \(\left(x + 1\right)^2\) because it is a perfect square trinomial.
Rewrite the rational expression using the factored denominator: \(\frac{x^2}{\left(x + 1\right)^2}\).
Set up the partial fraction decomposition form for a repeated linear factor \(\left(x + 1\right)^2\). The general form is: \(\frac{A}{x + 1} + \frac{B}{\left(x + 1\right)^2}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the common denominator \(\left(x + 1\right)^2\) to clear the fractions: \(x^2 = A(x + 1) + B\).
Expand the right side and collect like terms: \(x^2 = A x + A + B\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations to solve for \(A\) and \(B\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is especially useful for integrating rational functions or simplifying complex expressions. It involves breaking down a fraction into components with denominators that are factors of the original denominator.
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Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. For example, x^2 + 2x + 1 factors to (x + 1)^2. Recognizing and factoring the denominator is essential in partial fraction decomposition to determine the form of the simpler fractions.
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Handling Repeated Factors in Denominators

When the denominator has repeated factors, such as (x + 1)^2, the partial fraction decomposition must include terms for each power of the repeated factor. For (x + 1)^2, the decomposition includes terms with denominators (x + 1) and (x + 1)^2, each with its own numerator to solve for.
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Related Practice
Textbook Question

Find the cofactor of each element in the second row of each matrix.

[12−123−2−141]\(\left\)[ \(\begin{matrix}\) 1 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

y=6x+x2y=6x+x^2

4x−y=−34x-y=-3

Textbook Question

Graph each inequality. x ≤ 3

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Textbook Question

Write the system of equations associated with each augmented matrix . Do not solve.

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Textbook Question

Find the values of the variables for which each statement is true, if possible.

[05x−13y+241z]=[0w6−130418]\(\left\)[ \(\begin{matrix}\) 0 & 5 & x \\ -1 & 3 & y + 2 \\ 4 & 1 & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 0 & w & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each system by elimination. In systems with fractions, first clear denominators.

4x + y = -23

x - 2y = -17

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