Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 33

Chapter 6, Problem 33

Write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

Verified step by step guidance

Identify the given rational expression: $\frac{x+4}{x^{2}(x^{2}+4)}$.

Recognize the factors in the denominator: $x^{2}$ is a repeated linear factor (since $x$ is linear, repeated twice), and $x^{2}+4$ is an irreducible quadratic factor.

Set up the partial fraction decomposition form. For the repeated linear factor $x^{2}$, include terms with denominators $x$ and $x^{2}$, and for the irreducible quadratic $x^{2}+4$, include a term with a linear numerator:

\[\frac{x+4}{x^{2}(x^{2}+4)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{Cx + D}{x^{2} + 4}.\]

Multiply both sides of the equation by the common denominator $x^{2}(x^{2}+4)$ to clear the denominators:

\[x + 4 = A \cdot x (x^{2} + 4) + B (x^{2} + 4) + (Cx + D) x^{2}.\]

Expand the right side and then collect like terms by powers of $x$. This will allow you to equate coefficients of corresponding powers of $x$ on both sides to form a system of equations to solve for $A$, $B$, $C$, and $D$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with simpler denominators. This technique is useful for integration and solving equations involving rational expressions.

Factoring Polynomials

Factoring polynomials means rewriting them as products of simpler polynomials. Recognizing factors like quadratic expressions and their irreducibility over the reals is crucial for setting up the correct form of partial fractions.

Recommended video:

Guided course

07:30

Introduction to Factoring Polynomials

Related Practice

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^2 + 4y^2 = 20 $\x$ + 2y = 6\(\end{cases}$

views

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)x + 2y $\leq$ 4 $\y$ $\geq$ x - 3\(\end{cases}$

Textbook Question

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. 3x - 2y = − 5 4x + y = 8

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^3 + y = 0 $\x$^2 - y = 0\(\end{cases}$

Textbook Question

Write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x²+ x +1)