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Chapter 9: Problem 605

For the following exercises, write the partial fraction decomposition. $$ \frac{-8 x-30}{x^{2}+10 x+25} $$

Short Answer

Expert verified
\( \frac{-8}{x+5} + \frac{10}{(x+5)^2} \).

Step by step solution

01

Recognize the Denominator

First, observe the denominator \( x^2 + 10x + 25 \), which appears to be a perfect square trinomial.
02

Factor the Denominator

Factor the quadratic expression \( x^2 + 10x + 25 \) as \((x + 5)^2\). This indicates that the original fraction can be decomposed using these factors.
03

Set Up the Partial Fraction Decomposition

The partial fraction decomposition for \( \frac{-8x - 30}{(x + 5)^2} \) is given by \( \frac{A}{x + 5} + \frac{B}{(x + 5)^2} \).
04

Clear the Denominator

Multiply both sides by \((x + 5)^2\) to eliminate the fractions: \(-8x - 30 = A(x + 5) + B\).
05

Expand and Collect Terms

Expand the right side: \(-8x - 30 = A(x + 5) + B = Ax + 5A + B\).
06

Equate the Coefficients

Match the coefficients from both sides: For \(x\), \(-8 = A\). For the constant term, \(-30 = 5A + B\).
07

Solve the System of Equations

Substitute \(A = -8\) into the second equation: \(-30 = 5(-8) + B\) \(-30 = -40 + B\) \(B = 10\).
08

Write the Partial Fraction Decomposition

Substitute \(A\) and \(B\) back into the partial fractions: \(\frac{-8}{x + 5} + \frac{10}{(x + 5)^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratics
Factoring quadratic equations is a critical step in many algebraic processes, including partial fraction decomposition. A quadratic equation usually takes the form \( ax^2 + bx + c \). In some cases, like \( x^2 + 10x + 25 \), it appears as a perfect square trinomial. This means it can be expressed in the form \( (x + d)^2 \). For our specific example:
  • The trinomial \( x^2 + 10x + 25 \) is factored as \( (x + 5)^2 \).
Recognizing this form allows us to simplify the problem significantly. Whenever you encounter a quadratic, first check to see if it's a perfect square as this simplifies the factoring process. If it isn't, explore other factoring methods such as completing the square or using the quadratic formula.
Partial Fractions
The process of partial fractions is about expressing a complex fraction as a sum of simpler fractions. This is particularly useful for integration and solving differential equations. By writing \( \frac{-8x - 30}{(x + 5)^2} \) as the sum \( \frac{A}{x + 5} + \frac{B}{(x + 5)^2} \), we break it into easier parts.Here's why it works:
  • Each part of the decomposition corresponds to a factor in the denominator, and each term needs to capture all the "views" of the factors.
  • In this case, \( \frac{-8x - 30}{(x + 5)^2} \) has a repeated linear factor, so we use two terms: one with \( x + 5 \) and one with \( (x + 5)^2 \).
This setup allows us to handle the equation by focusing on simpler pieces. Once the base form of the partial fraction decomposition is set up, the next step is to find the values of \( A \) and \( B \) by clearing the denominators.
Solving Equations
Solving the resulting equations in partial fraction decomposition involves algebraic manipulation until each coefficient can be clearly defined. After setting up the partial fractions:
  • The equation \( -8x - 30 = A(x+5) + B \) is formed by removing the denominator.
Next, expand and align coefficients from both sides:
  • For \( x \), compare coefficients to find \( A \).
  • For constant terms, solve a straightforward equation to determine \( B \).
For example:- From \( -8 = A \), we determine \( A \) directly.- Substitute \( A \) into \( -30 = 5A + B \) to solve for \( B \), finding \( B = 10 \).Finally, substitute these values back into the decomposition to conclude the problem.

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Most popular questions from this chapter

For the following exercises, solve using a system of linear equations. A factory producing cell phones has the following cost and revenue functions: \(C(x)=x^{2}+75 x+2,688\) and \(R(x)=x^{2}+160 x .\) What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

For the following exercises, use substitution to solve the system of equations. $$ \begin{array}{l}{10 x+5 y=-5} \\ {3 x-2 y=-12}\end{array} $$

For the following exercises, use addition to solve the system of equations. $$ \begin{aligned} 3 x+4 y &=2 \\ 9 x+12 y &=3 \end{aligned} $$

For the following exercises, solve the system of three equations using substitution or addition. $$ \begin{aligned} x+y+z &=1 \\ 2 x+2 y+2 z &=1 \\ 3 x+3 y &=2 \end{aligned} $$

For the following exercises, use Cramer's Rule to solve the systems of equations. $$ \begin{array}{l}{0.1 x+0.1 y-0.1 z=-1.2} \\ {0.1 x-0.2 y+0.4 z=-1.2} \\ {0.5 x-0.3 y+0.8 z=-5.9}\end{array} $$
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